# Gröbner-Shirshov bases and linear bases for free multi-operated algebras
over algebras with applications to differential Rota-Baxter algebras and
integro-differential algebras

Zuan Liu, Zihao Qi, Yufei Qin and Guodong Zhou  Zuan Liu, Yufei Qin and
Guodong Zhou, School of Mathematical Sciences, Shanghai Key Laboratory of
PMMP, East China Normal University, Shanghai 200241, China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Zihao Qi, School of Mathematical
Sciences, Fudan University, Shanghai 200433, China<EMAIL_ADDRESS>
###### Abstract.

Quite much recent studies has been attracted to the operated algebra since it
unifies various notions such as the differential algebra and the Rota-Baxter
algebra. An $\Omega$-operated algebra is a an (associative) algebra equipped
with a set $\Omega$ of linear operators which might satisfy certain operator
identities such as the Leibniz rule. A free $\Omega$-operated algebra $B$ can
be generated on an algebra $A$ similar to a free algebra generated on a set.
If $A$ has a Gröbner-Shirshov basis $G$ and if the linear operators $\Omega$
satisfy a set $\Phi$ of operator identities, it is natural to ask when the
union $G\cup\Phi$ is a Gröbner-Shirshov basis of $B$. A previous work answers
this question affirmatively under a mild condition, and thereby obtains a
canonical linear basis of $B$.

In this paper, we answer this question in the general case of multiple linear
operators. As applications we get operated Gröbner-Shirshov bases for free
differential Rota-Baxter algebras and free integro-differential algebras over
algebras as well as their linear bases. One of the key technical difficulties
is to introduce new monomial orders for the case of two operators, which might
be of independent interest.

###### 2010 Mathematics Subject Classification:

16Z10 03C05 08B20 12H05 16S10 17B38

###### Contents

  1. 0.1 Operated GS basis theory: from a single operator to multiple operators
  2. 0.2 Free operated algebras over algebras
  3. 0.3 Differential Rota-Baxter algebras and integro-differential algebras
  4. 0.4 Outline of the paper
  5. 1 New monomial orders on free multi-operated semigroups and monoids
    1. 1.1 Free objects in the multi-operated setup
    2. 1.2 Free multi-operated semigroups and monoids
    3. 1.3 Monomial orders
  6. 2 Operator polynomial identities and multi-operated GS bases
    1. 2.1 Operator polynomial identities
    2. 2.2 Multi-operated GS bases for $\Phi$-algebras
    3. 2.3 Multi-operated GS basis for free $\Phi$-algebras over algebras
  7. 3 Free differential Rota-Baxter algebras over algebras
    1. 3.1  Case of nonunital algebras with $\lambda\neq 0$
    2. 3.2  Case of nonunital algebras with $\lambda=0$
    3. 3.3  Case of unital algebras
  8. 4 Free integro-differential algebras over algebras
    1. 4.1  Case of nonunital algebras with $\lambda\neq 0$
    2. 4.2  Case of nonunital algebras with $\lambda=0$
    3. 4.3  Case of unital algebras
    4. 4.4 Differential Rota-Baxter algebras vs integro-differential algebras

## Introduction

This paper extends the results of [17] to algebras endowed with several
operators, with applications to differential Rota-Baxter algebras and integro-
differential algebras.

###  0.1. Operated GS basis theory: from a single operator to multiple
operators

Since its introduction by Shirshov [20] and Buchberger [4] in the sixties of
last century, Gröbner-Shirshov (=GS) basis theory has become one of the main
tools of computational algebra; see for instance [10, 1, 3]. In order to deal
with algebras endowed with operators, Guo and his coauthors introduced a GS
basis theory in a series of papers [11, 23, 15, 6] (see also [2]) with the
goal to attack Rota’s program [19] to classify “interesting” operators on
algebras. Guo et al. considered operators satisfying some polynomial
identities, hence called operated polynomial identities (aka. OPIs) [11, 23,
15, 6]. Via GS basis theory and the somewhat equivalent theory: rewriting
systems, they could define when OPIs are GS. They are mainly interested into
two classes of OPIs: differential type OPIs and Rota-Baxter type OPIs, which
are carefully studied in [15, 23, 6]. For the state of art, we refer the
reader to the survey paper [8] and for recent development, see [22, 12, 17,
21].

In these papers [11, 23, 15, 6], the operated GS theory and hence Rota’s
classification program have been carried out only for algebras endowed with a
single operator. It would be very interesting to carry out further Rota’s
program for the general case of multiple linear operators.

The paper [2] contains a first step of this program by developing the GS basis
theory in this generalised setup. We will review and update the GS basis
theory in the multi-operated setup in Section 2.

Another direction is to generalise from operated algebras over a base field to
operated algebras over a base ring. While previous papers [17, 18] considered
this aspect for single operator case, this paper is aimed to deal with this
aspect for multiple linear operator case. In particular, some new monomial
orders for the two operator case will be constructed which enable us to study
operated GS bases for free operated algebras generated by algebras, while it
seems that the monomial orders appeared in previous papers can be applied
directly when the base ring is not a field any more.

###  0.2. Free operated algebras over algebras

Recently, there is a need to develop free operated algebras satisfying some
OPIs over a fixed algebras and construct GS bases and linear bases for these
free algebras as long as a GS basis is known for the given algebra. Ebrahimi-
Fard and Guo [5] used rooted trees and forests to give explicit constructions
of free noncommutative Rota-Baxter algebras on modules and sets; Lei and Guo
[16] constructed the linear basis of free Nijenhuis algebras over associative
algebras; Guo and Li [12] gave a linear basis of the free differential algebra
over associative algebras by introducing the notion of differential GS bases.

In a previous paper [17], the authors considered a question which can be
roughly stated as follows:

######  Question 0.1.

Given a (unital or nonunital) algebra $A$ with a GS basis $G$ and a set $\Phi$
of OPIs, assume that these OPIs $\Phi$ are GS in the sense of [2, 15, 23, 6].
Let $B$ be the free operated algebra satisfying $\Phi$ over $A$. When will
$\Phi\cup G$ be a GS basis for $B$?

They answer this question in the affirmative under a mild condition in [17,
Theorem 5.9]. When this condition is satisfied, $\Phi\cup G$ is a GS basis for
$B$ and as a consequence, we also get a linear basis of $B$. This result has
been applied to all Rota-Baxter type OPIs, a class of differential type OPIs,
averaging OPIs and Reynolds OPI in [17]. It was also applied to differential
type OPIs by introducing some new monomial orders [18].

In this paper, we consider a similar question for multi-operated algebras.

Let $\Omega$ be a nonempty set which will be the index set of operators.
Algebras endowed with operators indexed by $\Omega$ will be called
$\Omega$-algebras. OPIs can be extended to the multi-operated setup and one
can introduce the notion of $\Omega$-GS for OPIs.

######  Question 0.2.

Let $\Phi$ be a set of OPIs of a set of operators indexed by $\Omega$. Let $A$
be a (unital) algebra together with a GS basis $G$. Assume that these OPIs
$\Phi$ are GS in the sense of Section 2. Let $B$ be the free $\Omega$-algebra
over $A$ such that the operators satisfy $\Phi$. When will $\Phi\cup G$ be an
$\Omega$-GS basis for $B$?

We extend the main result of [17] to multi-operated cases; see Theorem 2.12
for unital algebras and Theorem 2.13 for nonunital algebras.

###  0.3. Differential Rota-Baxter algebras and integro-differential algebras

The main motivation of this paper comes, in fact, from differential Rota-
Baxter algebras and integro-differential algebras.

Differential Rota-Baxter algebras were introduced by Guo and Keigher [13]
which reflect the relation between the differential operator and the integral
operator as in the First Fundamental Theorem of Calculus. Free differential
Rota-Baxter algebras were constructed by using various tools including
angularly decorated rooted forests and GS basis theory [13, 2].

Integro-differential algebras (of zero weight) were defined for the algebraic
study of boundary problems for linear systems of linear ordinary differential
equations. Guo, Regensburger and Rosenkranz [14] introduced Integro-
differential algebras with weight. Free objects and their linear bases were
constructed by using GS basis theory [14, 9, 7]

The main goal of this paper is to study free differential Rota-Baxter algebras
and free integro-differential algebras over algebras from the viewpoint of
operated GS basis theory. In particular, when the base algebra is reduced to
${\bf k}$, our results also give GS bases and linear bases for free
differential Rota-Baxter algebras and free integro-differential algebras.

However, the original monomial orders used in [2, 14, 9, 7] do not satisfy the
hypothesis in Theorems 2.12 and 2.13 for free multi-operated algebras over
algebras, and we have to introduce a new monomial order
$\leq_{\operatorname{PD}}$ (resp. $\leq_{\operatorname{uPD}}$) to overcome the
problem; see Section 1.3.

In contrast to the use different monomial orders while dealing with free
differential Rota-Baxter algebras and free integro-differential algebras in
[2] and [7] respectively, we will demonstrate that our monomial ordering
$\leq_{\operatorname{PD}}$ can be applied to both types of algebras
simultaneously, as we shall see in Sections 3 and 4. Moreover, since the case
of the unital algebras was not discussed in [2], this aspect is addressed in
Subsection 3.3 by using our monomial order $\leq_{\operatorname{uPD}}$.

###  0.4. Outline of the paper

This paper is organized as follows.

The first section contains remainder on free objects in multi-operated setting
and on the construction of free $\Omega$-semigroups and related structures,
and introduces some new monomial orders for the case of two operators, which
will be the key technical tool of this paper.

In the second section, we recall the theory of GS bases for the multi-operated
setting. After introducing OPIs, GS property for OPIs and $\Omega$-GS bases
for multi-operated algebras are defined; after giving some facts about free
multi-operated $\Phi$-algebras on algebras, answers to Question 0.2 are
presented.

In the third section, multi-operated GS bases and linear bases for free
differential Rota-Baxter algebras on algebras are studied and the fourth
section contains our investigation for free integro-differential algebras on
algebras.

Notation: Throughout this paper, ${\mathbf{k}}$ denotes a base field. All the
vector spaces and algebras are over ${\mathbf{k}}$.

##  1\. New monomial orders on free multi-operated semigroups and monoids

In this section, we recall free objects in multi-operated setting and the
construction of free $\Omega$-semigroups and related structures, and define
two new monomial orders $\leq_{\operatorname{PD}}$ and
$\leq_{\operatorname{uPD}}$ on free multi-operated semigroups and monoids. The
main results of this paper will highly depend on these new monomial orders.

For a set $Z$, denote by ${\mathbf{k}}Z$ (resp. ${\mathcal{S}}(Z)$,
${\mathcal{M}}(Z)$) the free ${\mathbf{k}}$-vector space (resp. free
semigroup, free monoid) generated by $Z$. Denote the category of sets (resp.
semigroups, monoids) by ${\mathfrak{Set}}$ (resp. ${\mathfrak{Sem}}$,
${\mathfrak{Mon}}$). Denote the categories of ${\mathbf{k}}$-algebras and
unital ${\mathbf{k}}$-algebras by ${\mathfrak{Alg}}$ and ${\mathfrak{uAlg}}$
respectively.

Throughout this section, let $\Omega$ be a nonempty set which will be the
index set of operators.

###  1.1. Free objects in the multi-operated setup

######  Definition 1.1.

An operated set with an operator index set $\Omega$ or simply an $\Omega$-set
is a set $S$ endowed with a family of maps $P_{\omega}:S\rightarrow S$ indexed
by $\omega\in\Omega$. The morphisms between $\Omega$-sets can be defined in
the obvious way. Denote the category of $\Omega$-sets by
${\Omega\text{-}\mathfrak{Set}}$.

Similarly, we can define $\Omega$-semigroups and $\Omega$-monoids. Their
categories are denoted by ${\Omega\text{-}\mathfrak{Sem}}$ and
${\Omega\text{-}\mathfrak{Mon}}$ respectively.

$\Omega$-vector spaces, nonunital or unital $\Omega$-algebras can be defined
in a similar way, except asking, moreover, that all the operators are
${\mathbf{k}}$-linear maps. Denote the category of $\Omega$-vector spaces,
(resp. nonunital $\Omega$-algebras, unital $\Omega$-algebras) by
${\Omega\text{-}\mathfrak{Vect}}$ (resp. ${\Omega\text{-}\mathfrak{Alg}}$,
${\Omega\text{-}\mathfrak{uAlg}}$) with obvious morphisms.

As in [17], there exists the following diagram of functors:

$\textstyle{{\Omega\text{-}\mathfrak{Vect}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega\text{-}\mathfrak{Alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega\text{-}\mathfrak{uAlg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega\text{-}\mathfrak{Set}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega\text{-}\mathfrak{Sem}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega\text{-}\mathfrak{Mon}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{Vect}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{Alg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{uAlg}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{Set}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{Sem}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{Mon}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

In this diagram, all functors from right to left, from below to above and from
southwest to northeast are the obvious forgetful functors. The other functors
are free object functors which are left adjoint to the forgetful functors.

Our notations for free object functors are analogous to those in [17]. For
instance, ${\mathcal{F}}^{{\Omega\text{-}\mathfrak{Alg}}}_{{\mathfrak{Alg}}}$
denotes the free object functor from the category of algebras to that of
nonunital $\Omega$-algebras.

We could give similar constructions of these free object functors as in
Sections 1-3 of [17]. However, as we don’t need the details, we will not
repeat them. The curious readers could consult [17] and extend the
constructions in [17] without essential difficulties.

###  1.2. Free multi-operated semigroups and monoids

Now we explain the construction of the free $\Omega$-semigroup generated by a
set $Z$.

For $\omega\in\Omega$, denote by $\lfloor Z\rfloor_{\omega}$ the set of all
formal elements $\lfloor z\rfloor_{\omega},z\in Z$ and put $\lfloor
Z\rfloor_{\Omega}=\sqcup_{\omega\in\Omega}\left\lfloor
Z\right\rfloor_{\omega}$. The inclusion into the first component
$Z\hookrightarrow Z\sqcup\lfloor Z\rfloor_{\Omega}$ induces an injective
semigroup homomorphism

$i_{0,1}:{\mathfrak{S}_{\Omega}}_{,0}(Z):={\mathcal{S}}(Z)\hookrightarrow{\mathfrak{S}_{\Omega}}_{,1}(Z):={\mathcal{S}}(Z\sqcup\lfloor
Z\rfloor_{\Omega}).$

For $n\geq 2$, assume that we have constructed
${\mathfrak{S}_{\Omega}}_{,n-2}(Z)$ and
${\mathfrak{S}_{\Omega}}_{,n-1}(Z)={\mathcal{S}}(Z\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n-2}(Z)\rfloor_{\Omega})$
endowed with an injective homomorphism of semigroups
$i_{n-2,n-1}:{\mathfrak{S}_{\Omega}}_{,n-2}(Z)\hookrightarrow{\mathfrak{S}_{\Omega}}_{,n-1}(Z).$
We define the semigroup

${\mathfrak{S}_{\Omega}}_{,n}(Z):={\mathcal{S}}(Z\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n-1}(Z)\rfloor_{\Omega})$

and the natural injection

${\rm Id}_{Z}\sqcup\lfloor
i_{n-2,n-1}\rfloor_{\Omega}:Z\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n-2}(Z)\rfloor_{\Omega}\hookrightarrow
Z\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n-1}(Z)\rfloor_{\Omega}$

induces an injective semigroup homomorphism

$i_{n-1,n}:{\mathfrak{S}_{\Omega}}_{,n-1}(Z)={\mathcal{S}}(Z\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n-2}(Z)\rfloor_{\Omega})\hookrightarrow{\mathfrak{S}_{\Omega}}_{,n}(Z)={\mathcal{S}}(Z\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n-1}(Z)\rfloor_{\Omega}).$

Define ${\mathfrak{S}_{\Omega}}(Z)=\varinjlim{\mathfrak{S}_{\Omega}}_{,n}(Z)$
and the maps sending $u\in{\mathfrak{S}_{\Omega}}_{,n}(Z)$ to $\left\lfloor
u\right\rfloor_{\omega}\in{\mathfrak{S}_{\Omega}}_{,n+1}(Z)$ induces a family
of operators $P_{\omega},\omega\in\Omega$ on ${\mathfrak{S}_{\Omega}}(Z)$.

The construction of the free $\Omega$-monoid ${\mathfrak{M}_{\Omega}}(M)$ over
a set $Z$ is similar, by just replacing ${\mathcal{S}}(Z)$ by
${\mathcal{M}}(Z)$ everywhere in the construction.

######  Remark 1.2.

We will use another construction of ${\mathfrak{M}_{\Omega}}(Z)$. In fact, add
some symbols $\lfloor 1\rfloor_{\Omega}=\\{\lfloor
1\rfloor_{\omega},\omega\in\Omega\\}$ to $Z$ and form
${\mathfrak{S}_{\Omega}}(Z\sqcup\lfloor 1\rfloor_{\Omega})$, then
${\mathfrak{M}_{\Omega}}(Z)$ can be obtained from
${\mathfrak{S}_{\Omega}}(Z\sqcup\lfloor 1\rfloor_{\Omega})$ by just adding the
empty word $1$.

It is easy to see that ${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$(resp.
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$) is the free nonunital (resp. unital)
$\Omega$-algebra generated by $Z$.

###  1.3. Monomial orders

In this subsection, we introduce some new monomial orders on free
$\Omega$-semigroups and free $\Omega$-monoids. We only consider the case of
two operators, say $\Omega=\\{P,D\\}$ as the main examples in mind are
differential Rota-Baxter algebras and integro-differential algebras following
the convention from [7].

We first recall the definitions of well orders and monomial orders.

######  Definition 1.3.

Let $Z$ be a nonempty set.

  * (a)

A preorder $\leq$ is a binary relation on $Z$ that is reflexive and
transitive, that is, for all $x,y,z\in Z$, we have

    * (i)

$x\leq x$; and

    * (ii)

if $x\leq y,y\leq z$, then $x\leq z$.

In the presence of a preoder $\leq$, we denote $x=_{Z}y$ if $x\leq y$ and
$x\geq y$; if $x\leq y$ but $x\neq y$, we write $x<y$ or $y>x$.

  * (b)

A pre-linear order $\leq$ on $Z$ is a preorder $\leq$ such that either $x\leq
y$ or $x\geq y$ for all $x,y\in Z$.

  * (c)

A linear order or a total order $\leq$ on $Z$ is a pre-linear order $\leq$
such that $\leq$ is symmetric, that is, $x\leq y$ and $y\leq x$ imply $x=y$.

  * (d)

A preorder $\leq$ on $Z$ is said to satisfy the descending chain condition, if
for each descending chain $x_{1}\geq x_{2}\geq x_{3}\geq\cdots$, there exists
$N\geq 1$ such that $x_{N}=_{Z}x_{N+1}=_{Z}\cdots$. A linear order satisfying
the descending chain condition is called a well order.

Before giving the definition of monomial orders, we need to introduce the
following notions generalising the case of one operator.

######  Definition 1.4.

Let $Z$ be a set and $\star$ a symbol not in $Z$.

  * (a)

Define ${\mathfrak{M}_{\Omega}^{\star}}(Z)$ to be the subset of
${\mathfrak{M}_{\Omega}}(Z\cup\star)$ consisting of elements with $\star$
occurring only once.

  * (b)

For $q\in{\mathfrak{M}_{\Omega}^{\star}}(Z)$ and
$u\in{\mathfrak{M}_{\Omega}}(Z)$, we define
$q|_{u}\in{\mathfrak{M}_{\Omega}}(Z)$ to be the element obtained by replacing
the symbol $\star$ in $q$ by $u$. In this case, we say $u$ is a subword of
$q|_{u}$.

  * (c)

For $q\in{\mathfrak{M}_{\Omega}^{\star}}(Z)$ and
$s=\sum_{i}c_{i}u_{i}\in{\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with
$c_{i}\in{\mathbf{k}}$ and $u_{i}\in{\mathfrak{M}_{\Omega}}(Z)$, we define

$q|_{s}:=\sum_{i}c_{i}q|_{u_{i}}.$

  * (d)

Define ${\mathfrak{S}_{\Omega}^{\star}}(Z)$ to be the subset of
${\mathfrak{S}_{\Omega}}(Z\cup\star)$ consisting of elements with $\star$
occurring only once. It is easy to see ${\mathfrak{S}_{\Omega}^{\star}}(Z)$ is
a subset of ${\mathfrak{M}_{\Omega}^{\star}}(Z)$, so we also have notations in
(a)-(c) for ${\mathfrak{S}_{\Omega}^{\star}}(Z)$ by restriction.

######  Definition 1.5.

Let $Z$ be a set.

  * (a)

A monomial order on ${\mathcal{S}}(Z)$ is a well-order $\leq$ on
${\mathcal{S}}(Z)$ such that

$u<v\Rightarrow uw<vw\text{ and }wu<wv\text{ for any
}u,v,w\in{\mathcal{S}}(Z);$

  * (a’)

a monomial order on ${\mathcal{M}}(Z)$ is a well-order $\leq$ on
${\mathcal{M}}(Z)$ such that

$u<v\Rightarrow wuz<wvz\text{ for any }u,v,w,z\in{\mathcal{M}}(Z);$

  * (b)

a monomial order on ${\mathfrak{S}_{\Omega}}(Z)$ is a well-order $\leq$ on
${\mathfrak{S}_{\Omega}}(Z)$ such that

$u<v\Rightarrow q|_{u}<q|_{v}\quad\text{for all
}u,v\in{\mathfrak{S}_{\Omega}}(Z)\text{ and
}q\in{\mathfrak{S}_{\Omega}^{\star}}(Z);$

  * (b’)

a monomial order on ${\mathfrak{M}_{\Omega}}(Z)$ is a well-order $\leq$ on
${\mathfrak{M}_{\Omega}}(Z)$ such that

$u<v\Rightarrow q|_{u}<q|_{v}\quad\text{for all
}u,v\in{\mathfrak{M}_{\Omega}}(Z)\text{ and
}q\in{\mathfrak{M}_{\Omega}^{\star}}(Z).$

Let us recall some known preorders.

######  Definition 1.6.

For two elements $u,v\in{\mathfrak{S}_{\Omega}}(Z)$,

  * (a)

define

$u\leq_{\operatorname{D}}v\Leftrightarrow\operatorname{deg}_{D}(u)\leq\operatorname{deg}_{D}(v),$

where the $D$-degree $\operatorname{deg}_{D}(u)$ of $u$ is the number of
occurrence of $\lfloor~{}\rfloor_{D}$ in $u$;

  * (b)

define

$u\leq_{\operatorname{P}}v\Leftrightarrow\operatorname{deg}_{P}(u)\leq\operatorname{deg}_{P}(v),$

where the $P$-degree $\operatorname{deg}_{P}(u)$ of $u$ is the number of
occurrence of $\lfloor~{}\rfloor_{P}$ in $u$;

  * (c)

define

$u\leq_{\operatorname{dZ}}v\Leftrightarrow\operatorname{deg}_{Z}(u)\leq\operatorname{deg}_{Z}(v),$

where the $Z$-degree $\operatorname{deg}_{Z}(u)$ is the number of elements of
$Z$ occurring in $u$ counting the repetitions;

######  Definition 1.7.

Let $Z$ be a set endowed with a well order $\leq_{Z}$. Introduce the degree-
lexicographical order $\leq_{\rm{dlex}}$ on ${\mathcal{S}}(Z)$ by imposing,
for any $u\neq v\in{\mathcal{S}}(Z)$, $u<_{\rm{dlex}}v$ if

  * (a)

either $\operatorname{deg}_{Z}(u)<\operatorname{deg}_{Z}(v)$, or

  * (b)

$\operatorname{deg}_{Z}(u)=\operatorname{deg}_{Z}(v)$, and $u=mu_{i}n$,
$v=mv_{i}n^{\prime}$ for some $m,n,n^{\prime}\in{\mathcal{M}}(Z)$ and
$u_{i},v_{i}\in Z$ with $u_{i}<_{Z}v_{i}$.

It is obvious that the degree-lexicographic order $\leq_{\mathrm{dlex}}$ on
${\mathcal{S}}(Z)$ is a well order .

We now define a preorder $\leq_{\operatorname{Dlex}}$ on
${\mathfrak{S}_{\Omega}}(Z)$, by the following recursion process:

  * (a)

For $u,v\in{\mathfrak{S}_{\Omega}}_{,0}(Z)={\mathcal{S}}(Z)$, define

$u\leq_{\operatorname{Dlex}_{0}}v\Leftrightarrow u\leq_{\mathrm{dlex}}v.$

  * (b)

Assume that we have constructed a well order $\leq_{\operatorname{Dlex}_{n}}$
on ${\mathfrak{S}_{\Omega}}_{,n}(Z)$ for $n\geq 0$ extending all
$\leq_{\operatorname{Dlex}_{i}}$ for any $0\leq i\leq n-1$. The well order
$\leq_{\operatorname{Dlex}_{n}}$ on ${\mathfrak{S}_{\Omega}}_{,n}(Z)$ induces
a well order on $\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{P}$ (resp.
$\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{D}$), by imposing $\lfloor
u\rfloor_{P}\leq\lfloor v\rfloor_{P}$ (resp. $\lfloor u\rfloor_{D}\leq\lfloor
v\rfloor_{D}$) whenever
$u\leq_{\operatorname{Dlex}_{n}}v\in{\mathfrak{S}_{\Omega}}_{,n}(Z)$. By
setting $u<v<w$ for all $u\in Z$,
$v\in\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{D}$, and
$w\in\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{P}$, we obtain a well
order on
$Z\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{P}\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{D}$.
Let $\leq_{\operatorname{Dlex}_{n+1}}$ be the degree lexicographic order on
${\mathfrak{S}_{\Omega}}_{,n+1}(Z)={\mathcal{S}}(Z\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{P}\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{D})$
induced by that on
$Z\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{P}\sqcup\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{D}$.

Obviously $\leq_{\operatorname{Dlex}_{n+1}}$ extends
$\leq_{\operatorname{Dlex}_{n}}$. By a limit process, we get a preorder on
${\mathfrak{S}_{\Omega}}(Z)$ which will be denoted by
$\leq_{\operatorname{Dlex}}$. As is readily seen, $\leq_{\operatorname{Dlex}}$
is a linear order.

######  Remark 1.8.

It is easy to see that the above construction of $\leq_{\operatorname{Dlex}}$
can be extended to the case of more than two operators.

In fact, for a given well order $\leq_{\Omega}$ in the index set $\Omega$, the
defining process of $\leq_{\operatorname{Dlex}}$ on
${\mathfrak{S}_{\Omega}}(Z)$ is the same as above except one detail in step
(b), where we need to put $u<v<w$ for all $u\in Z$,
$v\in\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{\omega_{1}}$ and
$w\in\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{\omega_{2}}$ with
$\omega_{1}\leq_{\Omega}\omega_{2}\in\Omega$.

######  Definition 1.9.

For any $u\in{\mathfrak{S}_{\Omega}}(Z)$, let $u_{1},\dots,u_{n}\in Z$ be all
the elements occurring in $u$ from left to right. If a right half bracket
$\rfloor_{D}$ locates in the gap between $u_{i}$ and $u_{i+1}$, where $1\leq
i<n$, the GD-degree of this right half bracket is defined to be $n-i$; if
there is a right half bracket $\rfloor_{D}$ appearing on the right of $u_{n}$,
we define the GD-degree of this half bracket to be $0$. We define the GD-
degree of $u$, denoted by $\deg_{GD}(u)$, to be the sum of the GD-degrees of
all the half right brackets in $u$.

For example, the GD-degrees of the half right brackets in $u=\left\lfloor
x\right\rfloor_{D}\left\lfloor y\right\rfloor_{D}$ with $x,y\in Z$ are
respectively 1 and 0 from left to right, so $\deg_{GD}(u)=1$ by definition.

For $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, define the GD-degree order
$\leq_{\mathrm{GD}}$ by

$u\leq_{\mathrm{GD}}v\Leftrightarrow\deg_{GD}(u)\leq\deg_{GD}(v).$

######  Definition 1.10.

For any $u\in{\mathfrak{S}_{\Omega}}(Z)$, let $u_{1},\dots,u_{n}\in Z$ be all
the elements occurring in $u$ from left to right. If there are $i$ elements in
$Z$ contained in a bracket $\lfloor~{}\rfloor_{P}$ , the GP-degree of this
bracket is defined to be $n-i$. We denote by $\deg_{GP}(u)$ the sum of the GP-
degree of all the brackets $\lfloor~{}\rfloor_{P}$ in $u$.

For example, the the GP-degrees of the brackets $\lfloor~{}\rfloor_{P}$ of
$u=\left\lfloor xy\right\rfloor_{P}\left\lfloor z\right\rfloor_{P}$ with
$x,y,z\in Z$ are respectively 1 and 2 from left to right, so $\deg_{GP}(u)=3$
by definition.

For $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, define the GD-degree order
$\leq_{\mathrm{GD}}$ by

$u\leq_{\mathrm{GP}}v\Leftrightarrow\deg_{GP}(u)\leq\deg_{GP}(v).$

It is easy to obtain the following lemma whose proof is thus omitted.

######  Lemma 1.11.

The orders $\leq_{\mathrm{D}}$, $\leq_{\mathrm{P}}$, $\leq_{\mathrm{dZ}}$,
$\leq_{\operatorname{GD}}$ and $\leq_{\mathrm{GP}}$ are pre-linear orders
satisfying the descending chain condition.

Combining all the orders above, we can now construct an order
$\leq_{\operatorname{PD}}$ of ${\mathfrak{S}_{\Omega}}(Z)$:

$u\leq_{\operatorname{PD}}v\Leftrightarrow\left\\{\begin{array}[]{lcl}u\leq_{\mathrm{D}}v,\text{or
}\\\ u=_{\mathrm{D}}v\text{ and }u\leq_{\mathrm{P}}v,\text{or }\\\
u=_{\mathrm{D}}v,u=_{\mathrm{P}}v\text{ and }u\leq_{\mathrm{dZ}}v,\text{or
}\\\ u=_{\mathrm{D}}v,u=_{\mathrm{P}}v,u=_{\mathrm{dZ}}v\text{ and
}u\leq_{\mathrm{GD}}v,\text{or }\\\
u=_{\mathrm{D}}v,u=_{\mathrm{P}}v,u=_{\mathrm{dZ}}v,u=_{\mathrm{GD}}v\text{
and }u\leq_{\mathrm{GP}}v,\text{or }\\\
u=_{\mathrm{D}}v,u=_{\mathrm{P}}v,u=_{\mathrm{dZ}}v,u=_{\mathrm{GD}}v,u=_{\mathrm{GP}}v\text{
and }u\leq_{\mathrm{Dlex}}v.\end{array}\right.$

To prove that the $\leq_{\operatorname{PD}}$ is a well-order, we need some
preparation.

######  Definition 1.12.

  * (a)

Given some preorders $\leq_{\alpha_{1}},\dots,\leq_{\alpha_{k}}$ on a set $Z$
with $k\geq 2$, introduce another preorder
$\leq_{\alpha_{1},\dots,\alpha_{k}}$ by imposing recursively

$u\leq_{\alpha_{1},\dots,\alpha_{k}}v\Leftrightarrow\left\\{\begin{array}[]{l}u<_{\alpha_{1}}v,\text{
or }\\\ u=_{\alpha_{1}}v\text{ and
}u\leq_{\alpha_{2},\dots,\alpha_{k}}v.\end{array}\right.$

  * (b)

Let $k\geq 2$ and let $\leq_{\alpha_{i}}$ be a pre-linear order on
$Z_{i},~{}1\leq i\leq k$. Define the lexicographical product order
$\leq_{\text{clex}}$ on the cartesian product $Z_{1}\times
Z_{2}\times\cdots\times Z_{k}$ by defining

$(x_{1},\cdots,x_{k})\leq_{\text{clex}}(y_{1},\cdots,y_{k})\Leftrightarrow\left\\{\begin{array}[]{l}x_{1}<_{\alpha_{1}}y_{1},\text{or
}\\\ x_{1}=_{Z_{1}}y_{1}\text{ and
}\left(x_{2},\cdots,x_{k}\right)\leq_{\rm{clex}}\left(y_{2},\cdots,y_{k}\right),\end{array}\right.$

where
$\left(x_{2},\cdots,x_{k}\right)\leq_{\rm{clex}}\left(y_{2},\cdots,y_{k}\right)$
is defined by induction, with the convention that $\leq_{\rm{clex}}$ is the
trivial relation when $k=1$.

######  Lemma 1.13 ([18, Lemma 1.7]).

  * (a)

For $k\geq 2$, let $\leq_{\alpha_{1}},\dots,\leq_{\alpha_{k-1}}$ be pre-linear
orders on $Z$, and $\leq_{\alpha_{k}}$ a linear order on $Z$. Then
$\leq_{\alpha_{1},\dots,\alpha_{k}}$ is a linear order on $Z$.

  * (b)

Let $\leq_{\alpha_{i}}$ be a well order on $Z_{i}$, $1\leq i\leq k$. Then the
lexicographical product order $\leq_{\rm{clex}}$ is a well order on the
cartesian product $Z_{1}\times Z_{2}\times\cdots\times Z_{k}$.

######  Proposition 1.14.

The order $\leq_{\operatorname{PD}}$ is a well order on
${\mathfrak{S}_{\Omega}}(Z)$.

###### Proof.

Since $\leq_{\mathrm{Dlex}}$ is a linear order, so is
$\leq_{\operatorname{PD}}$ by Lemma 1.11 and Lemma 1.13(a).

It suffices to verify that $\leq_{\operatorname{PD}}$ satisfies the descending
chain condition. Let

$v_{1}\geq_{\operatorname{PD}}v_{2}\geq_{\operatorname{PD}}v_{3}\geq_{\operatorname{PD}}\cdots\in{\mathfrak{S}_{\Omega}}(Z)$

be a descending chain. By Lemma 1.11, there exist $N\geq 1$ such that

$\deg_{D}(v_{N})=\deg_{D}(v_{N+1})=\deg_{D}(v_{N+2})=\cdots=:k,$
$\deg_{P}(v_{N})=\deg_{P}(v_{N+1})=\deg_{P}(v_{N+2})=\cdots=:p,$
$\deg_{Z}(v_{N})=\deg_{Z}(v_{N+1})=\deg_{Z}(v_{N+2})=\cdots$
$\deg_{GD}(v_{N})=\deg_{GD}(v_{N+1})=\deg_{GD}(v_{N+2})=\cdots,$

and

$\deg_{GP}(v_{N})=\deg_{GP}(v_{N+1})=\deg_{GP}(v_{N+2})=\cdots.$

Thus all $v_{i}$ with $i\geq N$ belong to ${\mathfrak{S}_{\Omega}}_{,k+p}(Z)$.
The restriction of the order $\leq_{\mathrm{Dlex}}$ to
${\mathfrak{S}_{\Omega}}_{,k+p}(Z)$ equals to the well order
$\leq_{\operatorname{Dlex}_{k+p}}$, which by definition satisfies the
descending chain condition, so the chain
$v_{1}\geq_{\operatorname{PD}}v_{2}\geq_{\operatorname{PD}}v_{3}\geq_{\operatorname{PD}}\cdots$
stabilizes after finite steps. ∎

######  Definition 1.15 ([23, Definition 5.6]).

A preorder $\leq_{\alpha}$ on ${\mathfrak{S}_{\Omega}}(Z)$ is called bracket
compatible (resp. left compatible, right compatible) if

$u\leq_{\alpha}v\Rightarrow\lfloor u\rfloor_{D}\leq_{\alpha}\lfloor
v\rfloor_{D}\text{ and }\lfloor u\rfloor_{P}\leq_{\alpha}\lfloor
v\rfloor_{P},\text{ (resp. }wu\leq_{\alpha}wv,~{}uw\leq_{\alpha}vw,\ \text{
for all }w\in{\mathfrak{S}_{\Omega}}(Z))\ $

######  Lemma 1.16 ([23, Lemma 5.7]).

A well order $\leq$ is a monomial order on ${\mathfrak{S}_{\Omega}}(Z)$ if and
only if $\leq$ is bracket compatible, left compatible and right compatible.

Now we can prove the main result of this section which is the main technical
point of this paper.

######  Theorem 1.17.

The well order $\leq_{\operatorname{PD}}$ is a monomial order on
${\mathfrak{S}_{\Omega}}(Z)$.

###### Proof.

Let $u\leq_{\operatorname{PD}}v$. It is obvious that preorders
$\leq_{\mathrm{D}}$, $\leq_{\mathrm{P}}$ and $\leq_{\mathrm{dZ}}$ are bracket
compatible, left compatible and right compatible. This solves the three cases
$u<_{\mathrm{D}}v$; $u=_{\mathrm{D}}v$, $u<_{{\rm dgp}}v$; $u=_{\mathrm{D}}v$,
$u=_{{\rm dgp}}v$ and $u<_{{\rm dgx}}v$.

If $u=_{\mathrm{D}}v$, $u=_{\mathrm{P}}v,u=_{\mathrm{dZ}}v$ and
$u<_{\mathrm{GD}}v$, obviously $\lfloor u\rfloor_{D}<_{\mathrm{GD}}\lfloor
v\rfloor_{D}$, $\lfloor u\rfloor_{P}<_{\mathrm{GD}}\lfloor v\rfloor_{P}$
$uw<_{\mathrm{GD}}vw$ and $wu<_{\mathrm{GD}}wv$ for
$w\in{\mathfrak{S}_{\Omega}}(Z)$. So $\lfloor
u\rfloor_{D}<_{\mathrm{PD}}\lfloor v\rfloor_{D}$, $\lfloor
u\rfloor_{P}<_{\mathrm{PD}}\lfloor v\rfloor_{P}$, $uw<_{\mathrm{PD}}vw$ and
$wu<_{\mathrm{PD}}wv$.

The case that $u=_{\mathrm{D}}v$, $u=_{\mathrm{P}}v,u=_{\mathrm{dZ}}v$,
$u=_{\mathrm{GD}}v$ and $u<_{\mathrm{GP}}v$ is similar to the above one.

It remains to consider the case that $u=_{\mathrm{D}}v$,
$u=_{\mathrm{P}}v,u=_{\mathrm{dZ}}v$, $u=_{\mathrm{GD}}v$, $u=_{\mathrm{GP}}v$
and $u<_{\mathrm{Dlex}}v$. Let $n\geq\deg_{D}(u),\deg_{P}(u)$. Since
$u,v\in{\mathfrak{S}_{\Omega}}_{,n}(Z)$, thus
$u\leq_{\operatorname{Dlex}_{n}}v$. By the fact that the restriction of
$\leq_{\operatorname{Dlex}_{n+1}}$ to
$\lfloor{\mathfrak{S}_{\Omega}}_{,n}(Z)\rfloor_{D}$ is induced by
$\leq_{\operatorname{Dlex}_{n}}$, we have $\lfloor
u\rfloor_{D}\leq_{\operatorname{Dlex}_{n+1}}\lfloor v\rfloor_{D}$, $\lfloor
u\rfloor_{D}\leq_{\operatorname{Dlex}}\lfloor v\rfloor_{D}$, and $\lfloor
u\rfloor_{D}\leq_{\operatorname{PD}}\lfloor v\rfloor_{D}$. Similarly $\lfloor
u\rfloor_{P}\leq_{\operatorname{PD}}\lfloor v\rfloor_{P}$. Let
$w\in{\mathfrak{S}_{\Omega}}_{,m}(Z)$. One can obtain
$uw\leq_{\operatorname{Dlex}_{r}}vw$ and $wu\leq_{\operatorname{Dlex}_{r}}wv$
for $r=\max\left\\{m,n\right\\}$, so $uw\leq_{\operatorname{PD}}vw$ and
$wu\leq_{\operatorname{PD}}wv$.

We are done. ∎

Now let’s move to the unital case. Now we extend $\leq_{\operatorname{PD}}$
from ${\mathfrak{S}_{\Omega}}(Z)$ to ${\mathfrak{M}_{\Omega}}(Z)$ by using
Remark 1.2.

######  Definition 1.18.

Let $Z$ be a set with a well order. Let $\dagger_{P}$ (resp. $\dagger_{D}$ )
be a symbol which is understood to be $\lfloor 1\rfloor_{P}$ (resp. $\lfloor
1\rfloor_{D}$) and write $Z^{\prime}=Z\sqcup\\{\dagger_{P},\dagger_{D}\\}$.
Consider the free operated semigroup ${\mathfrak{S}_{\Omega}}(Z^{\prime})$
over the set $Z^{\prime}$. The well order on $Z$ extends to a well order
$\leq$ on $Z^{\prime}$ by setting $\dagger_{P}>z>\dagger_{D}$, for any $z\in
Z$. Besides, we impose $\operatorname{deg}_{P}(\dagger_{P})=1$ and
$\deg_{GP}(\dagger_{P})=0$. Then the monomial order $\leq_{\operatorname{PD}}$
on ${\mathfrak{S}_{\Omega}}(Z^{\prime})$ induces a well order
$\leq_{\operatorname{uPD}}$ on
${\mathfrak{M}_{\Omega}}(Z)={\mathfrak{S}_{\Omega}}(Z^{\prime})\sqcup\\{1\\}$
(in which $\lfloor 1\rfloor_{P}$ and $\lfloor 1\rfloor_{D}$ is identified with
$\dagger_{P}$ and $\dagger_{D}$ respectively), by setting
$u>_{\operatorname{uPD}}1$ for any $u\in{\mathfrak{S}_{\Omega}}(Z^{\prime})$.

######  Theorem 1.19.

The well order $\leq_{\operatorname{uPD}}$ is a monomial order on
${\mathfrak{M}_{\Omega}}(Z)$.

###### Proof.

Obviously, the well order $\leq_{\operatorname{uPD}}$ is bracket compatible on
${\mathfrak{M}_{\Omega}}(Z)\backslash\\{1\\}$. Let
$x\in{\mathfrak{M}_{\Omega}}(Z)\backslash\\{1\\}$. By definition,
$x>_{\operatorname{uPD}}1$. We have $\lfloor
x\rfloor_{P}>_{\mathrm{Dlex}}\lfloor 1\rfloor_{P}$ which implies $\lfloor
x\rfloor_{P}>_{\operatorname{uPD}}\dagger_{P}$. It is ready to see that
$\lfloor
x\rfloor_{D}>_{\operatorname{uPD}}x>_{\operatorname{uPD}}\dagger_{D}$. Thus
$\leq_{\operatorname{uPD}}$ is bracket compatible.

Clearly, $\leq_{\operatorname{uPD}}$ is left and right compatible. ∎

We record several important conclusions which will be useful later.

######  Proposition 1.20.

For any $u,v\in{\mathfrak{M}_{\Omega}}(Z)\backslash\\{1\\}$, we have

  * (a)

$\lfloor u\rfloor_{P}\lfloor 1\rfloor_{P}>_{\operatorname{uPD}}\lfloor
u\lfloor 1\rfloor_{P}\rfloor_{P}\geq_{\operatorname{uPD}}\lfloor\lfloor
u\rfloor_{P}\rfloor_{P}$,

  * (b)

$\lfloor 1\rfloor_{P}\lfloor v\rfloor_{P}>_{\operatorname{uPD}}\lfloor\lfloor
v\rfloor_{P}\rfloor_{P}\geq_{\operatorname{uPD}}\lfloor\lfloor
1\rfloor_{P}v\rfloor_{P}$,

  * (c)

$\lfloor 1\rfloor_{P}\lfloor 1\rfloor_{P}>_{\operatorname{uPD}}\lfloor\lfloor
1\rfloor_{P}\rfloor_{P}$,

  * (d)

$\lfloor 1\rfloor_{P}\lfloor v\rfloor_{D}>_{\operatorname{uPD}}\lfloor\lfloor
1\rfloor_{P}v\rfloor_{D}$,

  * (e)

$\lfloor u\rfloor_{D}\lfloor 1\rfloor_{P}>_{\operatorname{uPD}}\lfloor
u\lfloor 1\rfloor_{P}\rfloor_{D}$.

###### Proof.

Let
$u,v\in{\mathfrak{M}_{\Omega}}(Z)\backslash\\{1\\}={\mathfrak{S}_{\Omega}}(Z^{\prime})$.

  * (a)

It is easy to see $\lfloor\lfloor u\rfloor_{P}\rfloor_{P}$ have lowest
$\deg_{Z^{\prime}}$ among $\lfloor u\rfloor_{P}\lfloor 1\rfloor_{P},\lfloor
u\lfloor 1\rfloor_{P}\rfloor_{P},\lfloor\lfloor u\rfloor_{P}\rfloor_{P}$, and
we also have $\deg_{GP}(\lfloor u\rfloor_{P}\lfloor
1\rfloor_{P})>\deg_{GP}(\lfloor u\lfloor 1\rfloor_{P}\rfloor_{P})$.

  * (b)

It is similar to $(a)$.

  * (c)

It follows from $\deg_{Z^{\prime}}(\lfloor 1\rfloor_{P}\lfloor
1\rfloor_{P})>\deg_{Z^{\prime}}(\lfloor\lfloor 1\rfloor_{P}\rfloor_{P})$.

  * (d)

It can be deduced from $\lfloor 1\rfloor_{P}\lfloor
v\rfloor_{D}>_{\operatorname{Dlex}}\lfloor\lfloor 1\rfloor_{P}v\rfloor_{D}$ by
Definition 1.7.

  * (e)

It holds because $\deg_{GD}(\lfloor u\rfloor_{D}\lfloor
1\rfloor_{P})>\deg_{GD}(\lfloor u\lfloor 1\rfloor_{P}\rfloor_{D})$.

∎

##  2\. Operator polynomial identities and multi-operated GS bases

In this section, we extend the theory of operated GS bases due to [2, 15, 23,
6] from the case of single operator to multiple operators case. The
presentation is essentially contained in [7].

###  2.1. Operator polynomial identities

In this subsection, we give some basic notions and facts related to operator
polynomial identities. Throughout this section, $X$ denotes a set.

######  Definition 2.1.

We call an element
$\phi(x_{1},\dots,x_{n})\in{\mathbf{k}}{\mathfrak{S}_{\Omega}}(X)$ (resp.
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(X)$) with $n\geq 1,x_{1},\dots,x_{n}\in
X$ an operated polynomial identity (aka OPI).

From now on, we always assume that OPIs are multilinear, that is, they are
linear in each $x_{i}$.

######  Definition 2.2.

Let $\phi(x_{1},\dots,x_{n})$ be an OPI. A (unital) $\Omega$-algebra
$A=(A,\\{P_{\omega}\\}_{\omega\in\Omega})$ is said to satisfy the OPI
$\phi(x_{1},\dots,x_{n})$ if $\phi(r_{1},\dots,r_{n})=0,$ for all
$r_{1},\dots,r_{n}\in A.$ In this case,
$(A,\\{P_{\omega}\\}_{\omega\in\Omega})$ is called a (unital) $\phi$-algebra.

Generally, for a family $\Phi$ of OPIs, we call a (unital) $\Omega$-algebra
$(A,\\{P_{\omega}\\}_{\omega\in\Omega})$ a (unital) $\Phi$-algebra if it is a
(unital) $\phi$-algebra for any $\phi\in\Phi$. Denote the category of
$\Phi$-algebras (resp. unital $\Phi$-algebras) by
$\Phi\text{-}{\mathfrak{Alg}}$ (resp. $\Phi\text{-}{\mathfrak{uAlg}}$).

######  Definition 2.3.

An $\Omega$-ideal of an $\Omega$-algebra is an ideal of the associative
algebra closed under the action of the operators. The $\Omega$-ideal generated
by a subset $S\subseteq A$ is denoted by $\left\langle
S\right\rangle_{{\Omega\text{-}\mathfrak{Alg}}}$ (resp. $\left\langle
S\right\rangle_{{\Omega\text{-}\mathfrak{uAlg}}}$).

Obviously the quotient of an $\Omega$-algebra (resp. unital $\Omega$-algebra)
by an $\Omega$-ideal is naturally an $\Omega$-algebra (resp. $\Omega$-unital
algebra).

From now on, $\Phi$ denotes a family of OPIs in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(X)$ or
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(X)$. For a set $Z$ and a subset $Y$ of
${\mathfrak{M}_{\Omega}}(Z)$, introduce the subset
$S_{\Phi}(Y)\subseteq{\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ to be

$S_{\Phi}(Y):=\\{\phi(u_{1},\dots,u_{k})\ |\ u_{1},\dots,u_{k}\in
Y,~{}\phi(x_{1},\dots,x_{k})\in\Phi\\}.$

###  2.2. Multi-operated GS bases for $\Phi$-algebras

In this subsection, operated GS basis theory is extended to algebras with
multiple operators following closely [2].

######  Definition 2.4.

Let $Z$ be a set, $\leq$ a linear order on ${\mathfrak{M}_{\Omega}}(Z)$ and
$f\in{\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$.

  * (a)

Let $f\notin{\mathbf{k}}$. The leading monomial of $f$ , denoted by $\bar{f}$,
is the largest monomial appearing in $f$. The leading coefficient of $f$ ,
denoted by $c_{f}$, is the coefficient of $\bar{f}$ in $f$. We call $f$ monic
with respect to $\leq$ if $c_{f}=1$.

  * (a’)

Let $f\in{\mathbf{k}}$ (including the case $f=0$). We define the leading
monomial of $f$ to be $1$ and the leading coefficient of $f$ to be $c_{f}=f$.

  * (b)

A subset $S\subseteq{\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ is called
monicized with respect to $\leq$, if each nonzero element of $S$ has leading
coefficient $1$.

Obviously, each subset $S\subseteq{\mathfrak{M}_{\Omega}}(Z)$ can be made
monicized if we divide each nonzero element by its leading coefficient.

We need another notation. Let $Z$ be a set. For
$u\in{\mathfrak{M}_{\Omega}}(Z)$ with $u\neq 1$, as $u$ can be uniquely
written as a product $u_{1}\cdots u_{n}$ with $u_{i}\in
Z\cup\left\lfloor{\mathfrak{M}_{\Omega}}(Z)\right\rfloor_{\Omega}$ for $1\leq
i\leq n$, call $n$ the breadth of $u$, denoted by $|u|$; for $u=1$, we define
$|u|=0$.

######  Definition 2.5.

Let $\leq$ be a monomial order on ${\mathfrak{S}_{\Omega}}(Z)$ (resp.
${\mathfrak{M}_{\Omega}}(Z)$) and
$f,g\in{\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ (resp.
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$) be monic.

  * (a)

If there are $p,u,v\in{\mathfrak{S}_{\Omega}}(Z)$ (resp.
${\mathfrak{M}_{\Omega}}(Z)$) such that $p=\bar{f}u=v\bar{g}$ with
max$\left\\{|\bar{f}|,|\bar{g}|\right\\}<\left|p\right|<|\bar{f}|+|\bar{g}|$,
we call

$\left(f,g\right)^{u,v}_{p}:=fu-vg$

the intersection composition of $f$ and $g$ with respect to $p$.

  * (b)

If there are $p\in{\mathfrak{S}_{\Omega}}(Z)$ (resp.
${\mathfrak{M}_{\Omega}}(Z)$) and $q\in{\mathfrak{S}_{\Omega}^{\star}}(Z)$
(resp. ${\mathfrak{M}_{\Omega}^{\star}}(Z)$) such that
$p=\bar{f}=q|_{\bar{g}}$, we call

$\left(f,g\right)^{q}_{p}:=f-q|_{g}$

the inclusion composition of $f$ and $g$ with respect to $p$.

######  Definition 2.6.

Let $Z$ be a set and $\leq$ a monomial order on ${\mathfrak{S}_{\Omega}}(Z)$
(resp. ${\mathfrak{M}_{\Omega}}(Z)$). Let
${\mathcal{G}}\subseteq{\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ (resp.
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$).

  * (a)

An element $f\in{\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ (resp.
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$) is called trivial modulo
$({\mathcal{G}},p)$ for $p\in{\mathfrak{S}_{\Omega}}(Z)$ (resp.
${\mathfrak{M}_{\Omega}}(Z)$) if

$f=\underset{i}{\sum}c_{i}q_{i}|_{s_{i}}\text{ with
}q_{i}|_{\bar{s}_{i}}<p\text{, where }c_{i}\in{\mathbf{k}},\
q_{i}\in{\mathfrak{S}_{\Omega}^{\star}}(Z)~{}(\text{resp.}~{}{\mathfrak{M}_{\Omega}^{\star}}(Z))\
\mathrm{and}\ s_{i}\in{\mathcal{G}}.$

If this is the case, we write

$(f,g)_{p}\equiv 0\bmod({\mathcal{G}},p).$

In general, for any $u,v\in{\mathfrak{S}_{\Omega}}(Z)$ (resp.
${\mathfrak{M}_{\Omega}}(Z)$), $u\equiv v\bmod({\mathcal{G}},p)$ means that
$u-v=\left.\sum c_{i}q_{i}\right|_{s_{i}}$, with $q_{i}|_{\bar{s}_{i}}<p$,
where $c_{i}\in{\mathbf{k}},\ q_{i}\in{\mathfrak{S}_{\Omega}^{\star}}(Z)$
(resp. ${\mathfrak{M}_{\Omega}^{\star}}(Z))$ and $s_{i}\in{\mathcal{G}}$.

  * (b)

The subset ${\mathcal{G}}$ is called a GS basis in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ (resp.
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$) with respect to $\leq$ if, for all
pairs $f,g\in{\mathcal{G}}$ monicized with respect to $\leq$, every
intersection composition of the form $\left(f,g\right)^{u,v}_{p}$ is trivial
modulo $({\mathcal{G}},p)$, and every inclusion composition of the form
$\left(f,g\right)^{q}_{p}$ is trivial modulo $({\mathcal{G}},p)$.

To distinguish from usual GS bases for associative algebras, from now on, we
shall rename GS bases in multi-operated contexts by $\Omega$-GS bases.

######  Theorem 2.7.

(Composition-Diamond Lemma) Let $Z$ be a set, $\leq$ a monomial order on
${\mathfrak{M}_{\Omega}}(Z)$ and
${\mathcal{G}}\subseteq{\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$. Then the
following conditions are equivalent:

  * (a)

${\mathcal{G}}$ is an $\Omega$-GS basis in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$.

  * (b)

Denote

${\rm
Irr}({\mathcal{G}}):={\mathfrak{M}_{\Omega}}(Z)\backslash\left\\{q|_{\overline{s}}~{}|~{}s\in{\mathcal{G}},\
\ q\in{\mathfrak{M}_{\Omega}^{\star}}(Z)\right\\}.$

As a ${\mathbf{k}}$-space,
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)={\mathbf{k}}{\rm
Irr}({\mathcal{G}})\oplus\left\langle{\mathcal{G}}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}$
and ${\rm Irr}({\mathcal{G}})$ is a ${\mathbf{k}}$-basis of
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)/\left\langle{\mathcal{G}}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}$.

######  Theorem 2.8.

(Composition-Diamond Lemma) Let $Z$ be a set, $\leq$ a monomial order on
${\mathfrak{S}_{\Omega}}(Z)$ and
${\mathcal{G}}\subseteq{\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$. Then the
following conditions are equivalent:

  * (a)

${\mathcal{G}}$ is an $\Omega$-GS basis in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$.

  * (b)

Denote

${\rm
Irr}({\mathcal{G}}):={\mathfrak{S}_{\Omega}}(Z)\backslash\left\\{q|_{\overline{s}}~{}|~{}s\in{\mathcal{G}},\
\ q\in{\mathfrak{S}_{\Omega}^{\star}}(Z)\right\\}.$

As a ${\mathbf{k}}$-space,
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)={\mathbf{k}}{\rm
Irr}({\mathcal{G}})\oplus\left\langle{\mathcal{G}}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$
and ${\rm Irr}({\mathcal{G}})$ is a ${\mathbf{k}}$-basis of
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)/\left\langle{\mathcal{G}}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$.

######  Definition 2.9 ([6, Definiton 2.21(a)]).

  * (a)

Let $\Phi\subseteq{\mathbf{k}}{\mathfrak{S}_{\Omega}}(X)$ be a family of OPIs.
Let $Z$ be a set and $\leq$ a monomial order on ${\mathfrak{S}_{\Omega}}(Z)$.
We call $\Phi$ $\Omega$-GS on ${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with
respect to $\leq$ if $S_{\Phi}({\mathfrak{S}_{\Omega}}(Z))$ is an $\Omega$-GS
basis in ${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to $\leq$.

  * (b)

Let $\Phi\subseteq{\mathbf{k}}{\mathfrak{M}_{\Omega}}(X)$ be a family of OPIs.
Let $Z$ be a set and $\leq$ a monomial order on ${\mathfrak{M}_{\Omega}}(Z)$.
We call $\Phi$ $\Omega$-GS on ${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with
respect to $\leq$ if $S_{\Phi}({\mathfrak{M}_{\Omega}}(Z))$ is an $\Omega$-GS
basis in ${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with respect to $\leq$.

###  2.3. Multi-operated GS basis for free $\Phi$-algebras over algebras

In this subsection, we consider multi-operated GS basis for free
$\Phi$-algebras over algebras and generalise the main result of [17] to multi-
operated cases.

We will use the following results without proof as they are counterparts in
multi-operated setup of [17, Propositions 4.8].

######  Proposition 2.10.

  * (a)

Let $\Phi\subset{\mathbf{k}}{\mathfrak{S}_{\Omega}}(X)$ and
$A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ an algebra. Then

${\mathcal{F}}_{{\mathfrak{Alg}}}^{\Phi\text{-}{\mathfrak{Alg}}}(A):={\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)/\left\langle
S_{\Phi}({\mathfrak{S}_{\Omega}}(Z))\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$

is the free $\Phi$-algebra generated by $A$.

  * (b)

Let $\Phi\subset{\mathbf{k}}{\mathfrak{M}_{\Omega}}(X)$ and
$A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ a unital algebra. Then

${\mathcal{F}}_{{\mathfrak{uAlg}}}^{\Phi\text{-}{\mathfrak{uAlg}}}(A):={\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)/\left\langle
S_{\Phi}({\mathfrak{M}_{\Omega}}(Z))\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}$

is the free unital $\Phi$-algebra over $A$.

As in [17], we consider the following question:

######  Question 2.11.

Let $A$ be a (unital) algebra together with a Gröbner-Shirshov basis $G$.
Assume that a set $\Phi$ of operated polynomial identities is $\Omega$-GS in
the sense of Definition 2.9. Considering the free (unital) $\Phi$-algebra $B$
over $A$, when will the union “$\Phi\cup G$” be a $\Omega$-GS basis for $B$?

It is surprising that the answer of the corresponding question given in [17]
can be generalised to multi-operated case without much modifications.

######  Theorem 2.12.

Let $X$ be a set and $\Phi\subseteq{\mathbf{k}}{\mathfrak{M}_{\Omega}}(X)$ a
system of OPIs. Let $A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ be a unital algebra
with generating set $Z$. Assume that $\Phi$ is $\Omega$-GS on $Z$ with respect
to a monomial order $\leq$ in ${\mathfrak{M}_{\Omega}}(Z)$ and that $G$ is a
GS basis of $I_{A}$ in ${\mathbf{k}}{\mathcal{M}}(Z)$ with respect to the
restriction of $\leq$ to ${\mathcal{M}}(Z)$.

Suppose that the leading monomial of any OPI $\phi(x_{1},\dots,x_{n})\in\Phi$
has no subword in ${\mathcal{M}}(X)\backslash X$, and that
$\phi(u_{1},\dots,u_{n})$ vanishes or its leading monomial is
$\overline{\phi}(u_{1},\dots,u_{n})$ for all
$u_{1},\dots,u_{n}\in{\mathfrak{M}_{\Omega}}(Z)$. Then
$S_{\Phi}({\mathfrak{M}_{\Omega}}(Z))\cup G$ is an $\Omega$-GS basis of
$\left\langle S_{\Phi}({\mathfrak{M}_{\Omega}}(Z))\cup
I_{A}\right\rangle_{{\Omega\text{-}\mathfrak{uAlg}}}$ in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with respect to $\leq$.

###### Proof.

The proof of [17, Theorem 5.9] carries verbatim over multi-operated case,
because it reveals that the key point is that the leading monomial of any OPI
$\phi(x_{1},\dots,x_{n})\in\Phi$ has no subword in ${\mathcal{M}}(X)\backslash
X$.

For details, see the proof of [17, Theorem 5.9]. ∎

There exists a nonunital version of the above result, which is also a multi-
operated version of [18, Theorem 2.15].

######  Theorem 2.13.

Let $X$ be a set and $\Phi\subseteq{\mathbf{k}}{\mathfrak{S}_{\Omega}}(X)$ a
system of OPIs. Let $A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ be an algebra with
generating set $Z$. Assume that $\Phi$ is $\Omega$-GS on $Z$ with respect to a
monomial order $\leq$ in ${\mathfrak{S}_{\Omega}}(Z)$ and that $G$ is a GS
basis of $I_{A}$ in ${\mathbf{k}}{\mathcal{S}}(Z)$ with respect to the
restriction of $\leq$ to ${\mathcal{S}}(Z)$.

Suppose that the leading monomial of any OPI $\phi(x_{1},\dots,x_{n})\in\Phi$
has no subword in ${\mathcal{S}}(X)\backslash X$, and that for all
$u_{1},\dots,u_{n}\in{\mathfrak{S}_{\Omega}}(Z)$, $\phi(u_{1},\dots,u_{n})$
vanishes or its leading monomial is $\overline{\phi}(u_{1},\dots,u_{n})$. Then
$S_{\Phi}({\mathfrak{S}_{\Omega}}(Z))\cup G$ is an $\Omega$-GS basis of
$\left\langle S_{\Phi}({\mathfrak{S}_{\Omega}}(Z))\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$ in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to $\leq$.

##  3\. Free differential Rota-Baxter algebras over algebras

In this section, we apply Theorems 2.12 and 2.13 to differential Rota-Baxter
algebras.

From now on, let $\Omega=\\{D,P\\}$, fix a set $X=\\{x,y\\}$ with two elements
such that variables in OPIs will take values in $X$. When talking about
algebras or reductions of OPIs, fix a set $Z$ and we understand that variables
in OPIs will be replaced by elements of ${\mathfrak{S}_{\Omega}}(Z)$ or
${\mathfrak{M}_{\Omega}}(Z)$.

We first recall the definition of differential Rota-Baxter algebras. We use
$D(~{})$ and $P(~{})$ instead of the linear operators $\lfloor~{}\rfloor_{D}$
and $\lfloor~{}\rfloor_{P}$.

######  Definition 3.1 ([7, Definition 2.1]).

Let $\lambda\in{\mathbf{k}}$ be fixed.

  * (a)

A (unital) differential ${\mathbf{k}}$-algebra of weight $\lambda$ (also
called a (unital) $\lambda$-differential ${\mathbf{k}}$-algebra) is a (unital)
associative ${\mathbf{k}}$-algebra $R$ together with a linear operator
$D:R\rightarrow R$ such that

$\quad D(uv)=D(u)v+uD(v)+\lambda D(u)D(v)\quad\text{ for all }u,v\in R;$

when $R$ has a unity $1$, it is asked that $D(1)=0$.

  * (b)

A Rota-Baxter ${\mathbf{k}}$-algebra of weight $\lambda$ is an associative
${\mathbf{k}}$-algebra $R$ together with a linear operator $P:R\rightarrow R$
such that

$P(u)P(v)=P(uP(v))+P(P(u)v)+\lambda P(uv)\quad\text{ for all }u,v\in R.$

  * (c)

A (unital) differential Rota-Baxter ${\mathbf{k}}$-algebra of weight $\lambda$
(also called a (unital) $\lambda$-differential Rota-Baxter
${\mathbf{k}}$-algebra) is a (unital) differential k-algebra $(R,D)$ of weight
$\lambda$ and a Rota-Baxter operator $P$ of weight $\lambda$ such that

$D\circ P=\text{ id }.$

When we consider free differential Rota-Baxter algebras on algebras, it is
disappointing to see that the traditional order (see [2]) would not meet the
condition of Theorems 2.12 and 2.13. This is the intention of the new monomial
orders $\leq_{\operatorname{PD}}$ and $\leq_{\operatorname{uPD}}$ introduced
in Section 1.3.

###  3.1.  Case of nonunital algebras with $\lambda\neq 0$

Assume in this subsection that $\lambda\neq 0$. Denote

  * (1)

$\phi_{1}(x,y)=P(x)P(y)-P(xP(y))-P(P(x)y)-\lambda P(xy)$,

  * (2)

$\phi_{2}(x,y)=D(x)D(y)+\lambda^{-1}D(x)y+\lambda^{-1}xD(y)-\lambda^{-1}D(xy)$,

  * (3)

$\phi_{3}(x)=D(P(x))-x$.

We first consider nonunital free differential Rota-Baxter algebras on
algebras.

######  Proposition 3.2.

For any $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, the leading monomials of
$\phi_{1}(u,v)$, $\phi_{2}(u,v)$ and $\phi_{3}(u)$ under
$\leq_{\operatorname{PD}}$ are respectively $P(u)P(v),D(u)D(v)$ and $D(P(u))$.

###### Proof.

Let $u_{1},\cdots,u_{n}$ and $v_{1},\cdots,v_{m}$ be all the elements of $Z$
occurring in $u$ and $v$ from left to right.

For $\phi_{1}(u,v)=P(u)P(v)-P(uP(v))-P(P(u)v)-\lambda P(uv)$, we have
$\deg_{P}(P(uv))$ is smaller than those of the other three terms, while the
$\deg_{D},\deg_{Z}$ and $\deg_{GD}$ of the other elements are the same. And
one can see

$\deg_{GP}(P(u)P(v))-\deg_{GP}(P(uP(v)))=m>0,$
$\deg_{GP}(P(u)P(v))-\deg_{GP}(P(P(u)v))=n>0,$

so the leading monomial of $\phi_{1}(u,v)$ is $P(u)P(v)$.

The statements about $\phi_{2}(u,v)$ and $\phi_{3}(u)$ are obvious by
comparing $\deg_{D}$. ∎

Now let

${\mathrm{\Phi}_{\mathsf{DRB}}}^{\prime}:=\left\\{\phi_{1}(x,y),\phi_{2}(x,y),\phi_{3}(x)\right\\}.$

However, ${\mathrm{\Phi}_{\mathsf{DRB}}}^{\prime}$ is not $\Omega$-GS in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

######  Example 3.3.

For $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, let

$\begin{array}[]{rcl}f&=&\phi_{2}(P(u),v)=D(P(u))D(v)+\lambda^{-1}D(P(u))v+\lambda^{-1}P(u)D(v)-\lambda^{-1}D(P(u)v),\\\
\ g&=&\phi_{3}(u)=D(P(u))-u,\\\ q&=&\star D(v),\\\
p&=&D(P(u))D(v)=\bar{f}=\left.q\right|_{\bar{g}}.\end{array}$

Then

$(f,g)_{p}^{q}=f-\left.q\right|_{g}\equiv\lambda^{-1}(P(u)D(v)-D(P(u)v)+uv+\lambda
uD(v)).$

Let

$\phi_{4}(x,y)=P(x)D(y)-D(P(x)y)+xy+\lambda xD(y).$

It is clear that the leading monomial of $\phi_{4}(u,v)$ is $P(u)D(v)$ with
respect to $\leq_{\operatorname{PD}}$ which cannot be reduced further.

######  Example 3.4.

For $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, let

$\begin{array}[]{rcl}f&=&\phi_{2}(u,P(v))=D(u)D(P(v))+\lambda^{-1}D(u)P(v)+\lambda^{-1}uD(P(v))-\lambda^{-1}D(uP(v)),\\\
g&=&\phi_{3}(v)=D(P(v))-v,\\\ q&=&D(u)\star,\\\
p&=&D(u)D(P(v))=\bar{f}=\left.q\right|_{\bar{g}}.\end{array}$

Then

$(f,g)_{p}^{q}=f-\left.q\right|_{g}\equiv\lambda^{-1}(D(u)P(v)-D(uP(v))+uv+\lambda
D(u)v).$

Let

$\phi_{5}(x,y)=D(x)P(y)-D(xP(y))+xy+\lambda D(x)y.$

It is clear that the leading monomial of $\phi_{5}(u,v)$ is $D(u)P(v)$ with
respect to $\leq_{\operatorname{PD}}$ which cannot be reduced further.

Now denote ${\mathrm{\Phi}_{\mathsf{DRB}}}$ to be the set of the following
OPIs:

  * (1)

$\phi_{1}(x,y)=P(x)P(y)-P(xP(y))-P(P(x)y)-\lambda P(xy)$,

  * (2)

$\phi_{2}(x,y)=D(x)D(y)+\lambda^{-1}D(x)y+\lambda^{-1}xD(y)-\lambda^{-1}D(xy)$,

  * (3)

$\phi_{3}(x)=D(P(x))-x$,

  * (4)

$\phi_{4}(x,y)=P(x)D(y)-D(P(x)y)+xy+\lambda xD(y)$,

  * (5)

$\phi_{5}(x,y)=D(x)P(y)-D(xP(y))+xy+\lambda D(x)y$.

It is obvious that $\left\langle
S_{{\mathrm{\Phi}_{\mathsf{DRB}}}^{\prime}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}=\left\langle
S_{{\mathrm{\Phi}_{\mathsf{DRB}}}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$
for each set $Z$.

Next we will show that ${\mathrm{\Phi}_{\mathsf{DRB}}}$ is $\Omega$-GS with
respect to $\leq_{\operatorname{PD}}$. Before that, we need the following
lemma to simplify our proof.

######  Lemma 3.5.

Let $\phi(x_{1},\dots,x_{n})$ and $\psi(y_{1},\dots,y_{m})$ be two OPIs. Let
$Z$ be a set. Suppose that, for any
$u_{1},\dots,u_{n},v_{1},\dots,v_{m}\in{\mathfrak{S}_{\Omega}}(Z)$, the
leading monomial of $\phi(u_{1},\dots,u_{n})$ is
$\bar{\phi}(u_{1},\dots,u_{n})$ and leading monomial of
$\psi(v_{1},\dots,v_{m})$ is $\bar{\psi}(v_{1},\dots,v_{m})$.

Now write $f=\phi(u_{1},\dots,u_{n})$ and $g=\psi(v_{1},\dots,v_{m})$ for
fixed $u_{1},\dots,u_{n}$, $v_{1},\dots,v_{m}\in{\mathfrak{S}_{\Omega}}(Z)$.
If there exists $i~{}(1\leq i\leq n)$ and
$r\in{\mathfrak{S}_{\Omega}}^{\star}(Z)$ such that
$u_{i}=\left.r\right|_{\bar{g}}$, then the inclusion composition
$(f,g)_{p}^{q}=f-q|_{g}$ with $p=\bar{f}$ and
$q=\bar{\phi}(u_{1},\dots,u_{i-1},r,u_{i+1},\dots,u_{n})$, is trivial modulo
$(S_{\\{\phi,\psi\\}}(Z),w)$. We call this type of inclusion composition as
complete inclusion composition.

###### Proof.

The assertion follows from

$\displaystyle(f,g)_{p}^{q}$ $\displaystyle=f-q|_{g}$
$\displaystyle=(\phi-\bar{\phi})(u_{1},\dots,u_{i-1},r|_{\bar{g}},u_{i+1},\dots,u_{n})-\bar{\phi}(u_{1},\dots,u_{i-1},r|_{g-\bar{g}},u_{i+1},\dots,u_{n})$
$\displaystyle=(\phi-\bar{\phi})(u_{1},\dots,u_{i-1},r|_{g},u_{i+1},\dots,u_{n})-\phi(u_{1},\dots,u_{i-1},r|_{g-\bar{g}},u_{i+1},\dots,u_{n}).$

∎

######  Remark 3.6.

Lemma 3.5 extends [6, Theorem 4.1(b)] to the case of multiple operators.

Now we can prove ${\mathrm{\Phi}_{\mathsf{DRB}}}$ is $\Omega$-GS.

######  Theorem 3.7.

${\mathrm{\Phi}_{\mathsf{DRB}}}$ is $\Omega$-GS in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

###### Proof.

We write $i\wedge j$ the composition of OPIs of $\phi_{i}$ and $\phi_{j}$,
which means $\phi_{i}$ lies on the left and $\phi_{j}$ lies on the right for
intersection composition or $\phi_{j}$ is included in $\phi_{i}$ for inclusion
composition. The ambiguities of all possible compositions in
${\mathrm{\Phi}_{\mathsf{DRB}}}$ are listed as below: for arbitrary
$u,v,w\in{\mathfrak{S}_{\Omega}}(Z)$ and
$q\in{\mathfrak{S}_{\Omega}}^{\star}(Z)$,

  * $1\wedge 1$

$\underline{P(u)P(v)P(w)}$,
$P\left(\left.q\right|_{P(u)P(v)}\right)P\left(w\right)$,
$P\left(u\right)P\left(\left.q\right|_{P(v)P(w)}\right)$

  * $1\wedge 2$

$P\left(\left.q\right|_{D(u)D(v)}\right)P\left(w\right)$,
$P\left(u\right)P\left(\left.q\right|_{D(u)D(v)}\right)$,

  * $1\wedge 3$

$P\left(\left.q\right|_{D(P(u))}\right)P\left(v\right)$,
$P\left(u\right)P\left(\left.q\right|_{D(P(v))}\right)$,

  * $1\wedge 4$

$\underline{P(u)P(v)D(w)}$,
$P\left(\left.q\right|_{P(u)D(v)}\right)P\left(w\right)$,
$P\left(u\right)P\left(\left.q\right|_{P(v)D(w)}\right)$,

  * $1\wedge 5$

$P\left(\left.q\right|_{D(u)P(v)}\right)P\left(w\right)$,
$P\left(u\right)P\left(\left.q\right|_{D(v)P(w)}\right)$,

  * $2\wedge 1$

$D\left(\left.q\right|_{P(u)(P(v)}\right)D\left(w\right)$,
$D\left(u\right)D\left(\left.q\right|_{P(v)P(w)}\right)$,

  * $2\wedge 2$

$\underline{D(u)D(v)D(w)}$ ,
$D\left(\left.q\right|_{D(u)D(v)}\right)D\left(w\right)$,
$D\left(u\right)D\left(\left.q\right|_{D(v)D(w)}\right)$,

  * $2\wedge 3$

$\underline{D(P(u))D(v)}$, $\underline{D(u)D(P(v))}$,
$D\left(\left.q\right|_{D(P(u))}\right)D\left(v\right)$,
$D\left(u\right)D\left(\left.q\right|_{D(P(v))}\right)$,

  * $2\wedge 4$

$D\left(\left.q\right|_{P(u)D(v)}\right)D\left(w\right)$,
$D\left(u\right)D\left(\left.q\right|_{P(v)D(w)}\right)$,

  * $2\wedge 5$

$\underline{D(u)D(v)P(w)}$,
$D\left(\left.q\right|_{D(u)P(v)}\right)D\left(w\right)$,
$D\left(u\right)D\left(\left.q\right|_{D(v)P(w)}\right)$,

  * $3\wedge 1$

$D\left(P\left(\left.q\right|_{P(u)P(v)}\right)\right)$,

  * $3\wedge 2$

$D\left(P\left(\left.q\right|_{D(u)D(v)}\right)\right)$,

  * $3\wedge 3$

$D\left(P\left(\left.q\right|_{D(P(u))}\right)\right)$,

  * $3\wedge 4$

$D\left(P\left(\left.q\right|_{P(u)D(v)}\right)\right)$,

  * $3\wedge 5$

$D\left(P\left(\left.q\right|_{D(u)P(v)}\right)\right)$,

  * $4\wedge 1$

$P\left(\left.q\right|_{P(u)P(v)}\right)D\left(w\right)$,
$P\left(u\right)D\left(\left.q\right|_{P(v)P(w)}\right)$,

  * $4\wedge 2$

$\underline{P(u)D(v)D(w)}$,
$P\left(\left.q\right|_{D(u)D(v)}\right)D\left(w\right)$,
$P\left(u\right)D\left(\left.q\right|_{D(v)D(w)}\right)$,

  * $4\wedge 3$

$\underline{P(u)D(P(v))}$,
$P\left(\left.q\right|_{D(P(u))}\right)D\left(v\right)$,
$P\left(u\right)D\left(\left.q\right|_{D(P(v))}\right)$,

  * $4\wedge 4$

$P\left(\left.q\right|_{P(u)D(v)}\right)D\left(w\right)$,
$P\left(u\right)D\left(\left.q\right|_{P(v)D(w)}\right)$,

  * $4\wedge 5$

$\underline{P(u)D(v)P(w)}$,
$P\left(\left.q\right|_{D(u)P(v)}\right)D\left(w\right)$,
$P\left(u\right)D\left(\left.q\right|_{D(v)P(w)}\right)$,

  * $5\wedge 1$

$\underline{D(u)P(v)P(w)}$,
$D\left(\left.q\right|_{P(u)P(v)}\right)P\left(w\right)$,
$D\left(u\right)P\left(\left.q\right|_{P(v)P(w)}\right)$,

  * $5\wedge 2$

$D\left(\left.q\right|_{D(u)D(v)}\right)P\left(w\right)$,
$D\left(u\right)P\left(\left.q\right|_{D(v)D(w)}\right)$,

  * $5\wedge 3$

$\underline{D(P(u))P(v)}$,
$D\left(\left.q\right|_{D(P(u))}\right)P\left(v\right)$,
$D\left(u\right)P\left(\left.q\right|_{D(P(v))}\right)$,

  * $5\wedge 4$

$\underline{D(u)P(v)D(w)}$,
$D\left(\left.q\right|_{P(u)D(v)}\right)P\left(w\right)$,
$D\left(u\right)P\left(\left.q\right|_{P(v)D(w)}\right)$,

  * $5\wedge 5$

$D\left(\left.q\right|_{D(u)P(v)}\right)P\left(w\right)$,
$D\left(u\right)P\left(\left.q\right|_{D(v)P(w)}\right)$.

Notice that all compositions above but the underlined ones can be dealt with
by Lemma 3.5. There remains to consider the underlined compositions. We only
give the complete proof for the case $5\wedge 1$, the other cases being
similar. For the case $5\wedge 1$, write $f=\phi_{5}(u,v)$, $g=\phi_{1}(v,w)$
and $p=D(u)P(v)P(w)$ . So we have

$\displaystyle(f,g)_{p}^{P(w),D(u)}$
$\displaystyle=-D(uP(v))P(w)+uvP(w)+\lambda D(u)vP(w)$
$\displaystyle\quad+D(u)P(vP(w))+D(u)P(P(v)w)+\lambda D(u)P(vw)$
$\displaystyle\equiv-D(uP(v)P(w))+uP(v)w+\lambda D(uP(v))w+uvP(w)+\lambda
D(u)vP(w)$ $\displaystyle\quad+D(uP(P(v)w))-uP(v)w-\lambda D(u)P(v)w$
$\displaystyle\quad+D(uP(vP(w)))-uvP(w)-\lambda D(u)vP(w)$
$\displaystyle\quad+\lambda D(uP(vw))-\lambda uvw-\lambda^{2}D(u)vw$
$\displaystyle\equiv-D(uP(v)P(w))+D(uP(vP(w)))+D(uP(P(v)w))+\lambda D(uP(vw))$
$\displaystyle\quad-\lambda D(u)P(v)w+\lambda D(uP(v))w-\lambda
uvw-\lambda^{2}D(u)vw$
$\displaystyle=-D(u\phi_{1}(v,w))-\lambda\phi_{5}(u,v)w$ $\displaystyle\equiv
0\bmod\left(S_{{\mathrm{\Phi}_{\mathsf{DRB}}}}(Z),p\right).$

We are done. ∎

######  Theorem 3.8.

Let $Z$ be a set, $A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ a nonunital
${\mathbf{k}}$-algebra. Then we have:

${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{DRB}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)={\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)/\left\langle
S_{{\mathrm{\Phi}_{\mathsf{DRB}}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}.$

Moreover, assume $I_{A}$ has a GS basis $G$ with respect to the degree-
lexicographical order $\leq_{\rm{dlex}}$. Then
$S_{{\mathrm{\Phi}_{\mathsf{DRB}}}}(Z)\cup G$ is an operated GS basis of
$\left\langle S_{{\mathrm{\Phi}_{\mathsf{DRB}}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$ in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

###### Proof.

Since the leading monomial in ${\mathrm{\Phi}_{\mathsf{DRB}}}$ has no subword
in ${\mathcal{S}}(X)\backslash X$, the result follows immediately from Theorem
3.7 and Theorem 2.13. ∎

As a consequence, we obtain a linear basis.

######  Theorem 3.9.

Let $Z$ be a set, $A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ a nonunital
${\mathbf{k}}$-algebra with a GS basis $G$ with respect to $\leq_{\rm{dlex}}$.
Then the set ${\rm Irr}(S_{{\mathrm{\Phi}_{\mathsf{DRB}}}}(Z)\cup G)$ which is
by definition the complement of

$\left\\{q|_{\bar{s}},q|_{P(u)P(v)},q|_{D(u)D(v)},q|_{D(P(u))},q|_{P(u)D(v)},q|_{D(u)P(v)},s\in
G,q\in{\mathfrak{S}_{\Omega}^{\star}}(Z),u,v\in{\mathfrak{S}_{\Omega}}(Z)\right\\}$

in ${\mathfrak{S}_{\Omega}}(Z)$ is a linear basis of the free nonunital
$\lambda$-differential Rota-Baxter algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{DRB}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
over $A$.

###### Proof.

It can be induced directly by Theorem 2.8. ∎

######  Remark 3.10.

Since the monomial order used in [2] does not satisfy the conditions of
Theorem 2.13, we have to make use of a new monomial order while treating free
differential Rota-Baxter algebras over an algebra. In fact, since the leading
monomials are different, even for free differential Rota-Baxter algebras over
a field, our monomial order will provide new operated GS basis and linear
basis.

###  3.2.  Case of nonunital algebras with $\lambda=0$

Now we consider nonunital free differential Rota-Baxter algebras on algebras
with $\lambda=0$. This case can be studied similarly to the case $\lambda\neq
0$, so we omit the details in this subsection.

Denote $\phi_{1}(x,y)$ with $\lambda=0$ by $\phi_{1}^{0}(x,y)$ . Let

$\phi_{2}^{0}(x,y)=D(x)y+xD(y)-D(xy).$

We also write $\phi_{3}^{0}(x)=\phi_{3}(x)$ for convenience.

######  Proposition 3.11.

For any $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, the leading monomials of
$\phi_{1}^{0}(u,v)$, $\phi_{2}^{0}(u,v)$ and $\phi_{3}^{0}(u)$ with respect to
$\leq_{\operatorname{PD}}$ are $P(u)P(v),D(u)v$ and $D(P(u))$ respectively.

Let

${\mathrm{\Phi}_{\mathsf{DRB}}^{0}}^{\prime}:=\left\\{~{}\phi_{1}^{0}(x,y),\phi_{2}^{0}(x,y),\phi_{3}^{0}(x)\right\\}.$

By the following example, one can see that
${\mathrm{\Phi}_{\mathsf{DRB}}^{0}}^{\prime}$ is not $\Omega$-GS in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

######  Example 3.12.

For $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, let

$\begin{array}[]{rcl}f&=&\phi_{2}^{0}(P(u),v)=D(P(u))v+P(u)D(v)-D(P(u)v),\\\ \
g&=&\phi_{3}^{0}(u)=D(P(u))-u,\\\ q&=&\star v,\\\
p&=&D(P(u))v=\bar{f}=\left.q\right|_{\bar{g}}.\end{array}$

Then

$(f,g)_{p}^{q}=f-\left.q\right|_{g}\equiv P(u)D(v)-D(P(u)v)+uv.$

Let

$\phi_{4}^{0}(x,y)=P(x)D(y)-D(P(x)y)+xy.$

It is clear that the leading monomial of $\phi_{4}^{0}(u,v)$ with
$u,v\in{\mathfrak{S}_{\Omega}}(Z)$ is $P(u)D(v)$ with respect to
$\leq_{\operatorname{PD}}$ which cannot be reduced further.

Now denote ${\mathrm{\Phi}_{\mathsf{DRB}}^{0}}$ to be the set of the following
OPIs:

  * (1)

$\phi_{1}^{0}(x,y)=P(x)P(y)-P(xP(y))-P(P(x)y)$,

  * (2)

$\phi_{2}^{0}(x,y)=D(x)y+xD(y)-D(xy)$,

  * (3)

$\phi_{3}^{0}(x)=D(P(x))-x$,

  * (4)

$\phi_{4}^{0}(x,y)=P(x)D(y)-D(P(x)y)+xy$.

It is obvious that $\left\langle
S_{{\mathrm{\Phi}_{\mathsf{DRB}}^{0}}^{\prime}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}=\left\langle
S_{{\mathrm{\Phi}_{\mathsf{DRB}}^{0}}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$
for arbitrary set $Z$.

Similar to the case $\lambda\neq 0$, it can be proved that
${\mathrm{\Phi}_{\mathsf{DRB}}^{0}}$ is $\Omega$-GS with respect to
$\leq_{\operatorname{PD}}$.

######  Remark 3.13.

Note that $\phi_{4}^{0}(x,y)$ is just $\phi_{4}(x,y)$ with $\lambda=0$, and
for $u,v\in{\mathfrak{S}_{\Omega}}(Z)$

$\phi_{2}^{0}(u,P(v))=D(u)P(v)+uD(P(v))-D(uP(v))\equiv D(u)P(v)+uv-D(uP(v)),$

which is exactly $\phi_{5}(u,v)$ with $\lambda=0$. So $\phi_{5}(x,y)$
($\lambda=0$) does not appear in ${\mathrm{\Phi}_{\mathsf{DRB}}^{0}}$.

######  Theorem 3.14.

${\mathrm{\Phi}_{\mathsf{DRB}}^{0}}$ is $\Omega$-GS in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

###### Proof.

As in the proof of Theorem 3.7, we write $i\wedge j$ the composition of OPIs
of $\phi_{i}$ and $\phi_{j}$. There are two kinds of ambiguities of all
possible compositions in ${\mathrm{\Phi}_{\mathsf{DRB}}^{0}}$. Since
$\phi_{1}^{0}(x,y)$, $\phi_{3}^{0}(x)$, and $\phi_{4}^{0}(x,y)$ have the same
leading monomials as in the case $\lambda\neq 0$, the corresponding
ambiguities $i\wedge j$ with $i,j\in\\{1,3,4\\}$ are the same in the proof of
Theorem 3.7. Since $\phi^{0}_{2}(x,y)$ has a different leading monomial, the
ambiguities of the case $i\wedge j$ with $i=2$ or $j=2$ are the following: for
arbitrary
$u,v,w\in{\mathfrak{S}_{\Omega}}(Z),q\in{\mathfrak{S}_{\Omega}}^{\star}(Z)$
and $s\in{\mathfrak{S}_{\Omega}}(Z)\text{ or }\emptyset$,

  * $1\wedge 2$

$P\left(\left.q\right|_{D(u)v}\right)P\left(w\right)$,
$P\left(u\right)P\left(\left.q\right|_{D(v)w}\right)$;

  * $2\wedge 1$

$D\left(\left.q\right|_{P(u)P(v)}\right)w$,
$D\left(u\right)\left.q\right|_{P(u)P(v)}$;

  * $2\wedge 2$

$\underline{D(u)sD(v)w}$, $D\left(\left.q\right|_{D(u)v}\right)w$,
$D\left(u\right)\left.q\right|_{D(v)w}$;

  * $2\wedge 3$

$\underline{D\left(P(u)\right)v}$, $D\left(\left.q\right|_{D(P(u))}\right)v$,
$D\left(u\right)\left.q\right|_{D(P(v))}$;

  * $2\wedge 4$

$\underline{D(u)sP(v)D(w)}$, $D\left(\left.q\right|_{P(u)D(v)}\right)w$,
$D\left(u\right)\left.q\right|_{P(v)D(w)}$;

  * $3\wedge 2$

$D\left(P\left(\left.q\right|_{D(u)v}\right)\right)$;

  * $4\wedge 2$

$\underline{P(u)D(v)w}$,
$P\left(\left.q\right|_{D(u)v}\right)D\left(w\right)$,
$P\left(u\right)D\left(\left.q\right|_{D(v)w}\right)$.

Almost all the cases can be treated similarly as in the proof of Theorem 3.7,
except a slight difference in the case $2\wedge 2$. In fact, let
$f=\phi_{2}^{0}(u,sD(v))$, $g=\phi_{2}^{0}(v,w)$ and $p=D(u)sD(v)w$. So we
have

$\displaystyle(f,g)_{p}^{D(u)s,D(w)}$
$\displaystyle=uD(sD(v))w-D(usD(v))w-D(u)svD(w)+D(u)sD(vw)$
$\displaystyle\equiv-usD(v)D(w)+uD(sD(v)w)+usD(v)D(w)-D(usD(v)w)$
$\displaystyle\quad+uD(svD(w))-D(usvD(w))-uD(sD(vw))+D(usD(vw))$
$\displaystyle=uD(s\phi_{2}^{0}(v,w))-D(us\phi_{2}^{0}(v,w))$
$\displaystyle\equiv
0\bmod\left(S_{{\mathrm{\Phi}_{\mathsf{DRB}}^{0}}}(Z),p\right).$

We are done. ∎

######  Theorem 3.15.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ a nonunital
${\mathbf{k}}$-algebra. Then we have:

${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{DRB}}^{0}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)={\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)/\left\langle
S_{{\mathrm{\Phi}_{\mathsf{DRB}}^{0}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}.$

Moreover, assume $I_{A}$ has a GS basis $G$ with respect to the degree-
lexicographical order $\leq_{\rm{dlex}}$. Then
$S_{{\mathrm{\Phi}_{\mathsf{DRB}}^{0}}}(Z)\cup G$ is an operated GS basis of
$\left\langle S_{{\mathrm{\Phi}_{\mathsf{DRB}}^{0}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$ in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

As a consequence, we obtain a linear basis.

######  Theorem 3.16.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ a nonunital
${\mathbf{k}}$-algebra with a GS basis $G$ with respect to $\leq_{\rm{dlex}}$.
Then the set ${\rm Irr}(S_{{\mathrm{\Phi}_{\mathsf{DRB}}^{0}}}(Z)\cup G)$
which is by definition the complement of

$\left\\{q|_{\bar{s}},q|_{P(u)P(v)},q|_{D(u)v},q|_{D(P(u))},q|_{P(u)D(v)},s\in
G,q\in{\mathfrak{S}_{\Omega}^{\star}}(Z),u,v\in{\mathfrak{S}_{\Omega}}(Z)\right\\}$

in ${\mathfrak{S}_{\Omega}}(Z)$ is a linear basis of the free nonunital
$0$-differential Rota-Baxter algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{DRB}}^{0}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
over $A$.

###  3.3.  Case of unital algebras

Now we consider unital differential Rota-Baxter algebras. Since the proofs are
similar to those in the previous subsections, we omit most of them. The study
still divided into cases of $\lambda\neq 0$ and $\lambda=0$.

When $\lambda\neq 0$, since unital differential Rota-Baxter algebras have the
condition $D(1)=0$, put ${\mathrm{\Phi}_{\mathsf{uDRB}}}$ to be the union of
${\mathrm{\Phi}_{\mathsf{DRB}}}$ with $\\{D(1)\\}$, but by abuse of notation,
in ${\mathrm{\Phi}_{\mathsf{DRB}}}$, $x,y$ take their values in
${\mathfrak{M}_{\Omega}}(Z)$ instead of ${\mathfrak{S}_{\Omega}}(Z)$.

######  Remark 3.17.

We have:

$\left\\{\begin{array}[]{lll}\phi_{2}(u,v)\equiv 0&\text{ when }u=1\text{ or
}v=1;\\\ \phi_{4}(u,v)\equiv-D(P(u))+u=-\phi_{3}(u)&\text{ when }v=1;\\\
\phi_{5}(u,v)\equiv-D(P(v))+v=-\phi_{3}(v)&\text{ when
}u=1.\end{array}\right.$

So adding of the unity $1$ will not produce new OPIs. Moreover, it is clear
that except the above cases, the leading monomials of OPIs in
${\mathrm{\Phi}_{\mathsf{DRB}}}$ are the same with respect to
$\leq_{\operatorname{PD}}$ and $\leq_{\operatorname{uPD}}$ by Proposition
1.20.

With similar proofs as in Subsection 3.1, we can prove the following results.

######  Theorem 3.18.

${\mathrm{\Phi}_{\mathsf{uDRB}}}$ is $\Omega$-GS in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{uPD}}$.

######  Theorem 3.19.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ a unital
${\mathbf{k}}$-algebra. Then we have:

${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{uDRB}}}\text{-}{\mathfrak{uAlg}}}_{{\mathfrak{uAlg}}}(A)={\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)/\left\langle
S_{{\mathrm{\Phi}_{\mathsf{uDRB}}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}.$

Moreover, assume $I_{A}$ has a GS basis $G$ with respect to the degree-
lexicographical order $\leq_{\rm{dlex}}$. Then
$S_{{\mathrm{\Phi}_{\mathsf{uDRB}}}}(Z)\cup G$ is an operated GS basis of
$\left\langle S_{{\mathrm{\Phi}_{\mathsf{uDRB}}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}$ in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{uPD}}$.

######  Theorem 3.20.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ a unital
${\mathbf{k}}$-algebra with a GS basis $G$ with respect to $\leq_{\rm{dlex}}$.
Then the set ${\rm Irr}(S_{{\mathrm{\Phi}_{\mathsf{uDRB}}}}(Z)\cup G)$ which
is by definition the complement of

$\left\\{q|_{\bar{s}},q|_{P(u)P(v)},q|_{D(u)D(v)},q|_{D(P(u))},q|_{P(u)D(v)},q|_{D(u)P(v)},q|_{D(1)},s\in
G,q\in{\mathfrak{M}_{\Omega}^{\star}}(Z),u,v\in{\mathfrak{M}_{\Omega}}(Z)\right\\}$

in ${\mathfrak{M}_{\Omega}}(Z)$ is a linear basis of the free unital
$\lambda$-differential Rota-Baxter algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{uDRB}}}\text{-}{\mathfrak{uAlg}}}_{{\mathfrak{uAlg}}}(A)$
over $A$.

When $\lambda=0$, denote
${\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}:={\mathrm{\Phi}_{\mathsf{DRB}}^{0}}$
(again by abuse of notation, ${\mathrm{\Phi}_{\mathsf{DRB}}^{0}}$ is
understood that $u,v$ take their values in ${\mathfrak{M}_{\Omega}}(X)$
instead of ${\mathfrak{S}_{\Omega}}(X)$).

######  Remark 3.21.

In ${\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}$, we have

$\phi_{2}^{0}(1,1)=D(1)+D(1)-D(1)=D(1),$

so it is not necessary to add $D(1)$ into
${\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}$.

Note that $\phi_{4}^{0}(u,1)\equiv-D(P(v))+v=-\phi_{3}^{0}(v)$, so adding the
unity $1$ will not induce any new OPI.

By using similar proofs in Subsection 3.2, one can show the following results.

######  Theorem 3.22.

${\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}$ is $\Omega$-GS in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with $\leq_{\operatorname{uPD}}$.

######  Theorem 3.23.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ a unital
${\mathbf{k}}$-algebra. Then we have:

${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}\text{-}{\mathfrak{uAlg}}}_{{\mathfrak{uAlg}}}(A)={\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)/\left\langle
S_{{\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}.$

Moreover, assume $I_{A}$ has a GS basis $G$ with respect to the degree-
lexicographical order $\leq_{\rm{dlex}}$. Then
$S_{{\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}}(Z)\cup G$ is an operated GS basis of
$\left\langle S_{{\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}$ in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{uPD}}$.

######  Theorem 3.24.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ a unital
${\mathbf{k}}$-algebra with a GS basis $G$ with respect to $\leq_{\rm{dlex}}$.
Then the set ${\rm Irr}(S_{{\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}}(Z)\cup G)$
which is by definition the complement of

$\left\\{q|_{\bar{s}},q|_{P(u)P(v)},q|_{D(u)v},q|_{D(P(u))},q|_{P(u)D(v)},s\in
G,q\in{\mathfrak{M}_{\Omega}^{\star}}(Z),u,v\in{\mathfrak{M}_{\Omega}}(Z)\right\\}$

in ${\mathfrak{M}_{\Omega}}(Z)$ is a linear basis of the free unital
$0$-differential Rota-Baxter algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}\text{-}{\mathfrak{uAlg}}}_{{\mathfrak{uAlg}}}(A)$
over $A$.

So far, we have completed the study of differential Rota-Baxter algebras.

##  4\. Free integro-differential algebras over algebras

In this section, we carry the study of GS bases of free integro-differential
algebras over algebras. It reveals that integro-differential algebras can be
investigated by using a method similar to differential Rota-Baxter algebras,
but the details are more difficult.

We first recall the definition of integro-differential algebras.

######  Definition 4.1.

Let $\lambda\in{\bf k}$. An integro-differential $\mathbf{k}$-algebra of
weight $\lambda$ (also called a $\lambda$-integro-differential
$\mathbf{k}$-algebra) is a differential $\mathbf{k}$-algebra $(R,d)$ of weight
$\lambda$ with a linear operator $P:R\rightarrow$ $R$ which satisfies (c) in
Definition 3.1:

$D\circ P=\text{ id },$

and such that

$\begin{array}[]{ll}P(D(u)P(v))=uP(v)-P(uv)-\lambda P(D(u)v)&\text{ for all
}u,v\in R,\\\ P(P(u)D(v))=P(u)v-P(uv)-\lambda P(uD(v))&\text{ for all }u,v\in
R.\end{array}$

Recall that

  * (1)

$\phi_{1}(x,y)=P(x)P(y)-P(xP(y))-P(P(x)y)-\lambda P(xy)$,

  * (2)

$\phi_{2}(x,y)=D(x)D(y)+\lambda^{-1}D(x)y+\lambda^{-1}xD(y)-\lambda^{-1}D(xy)$,

  * (3)

$\phi_{3}(x)=D(P(x))-x$,

  * (4)

$\phi_{4}(x,y)=P(x)D(y)-D(P(x)y)+xy+\lambda xD(y)$,

  * (5)

$\phi_{5}(x,y)=D(x)P(y)-D(xP(y))+xy+\lambda D(x)y$,

and denote

  * (6)

$\phi_{6}(x,y)=P(D(x)P(y))-xP(y)+P(xy)+\lambda P(D(x)y)$,

  * (7)

$\phi_{7}(x,y)=P(P(x)D(y))-P(x)y+P(xy)+\lambda P(xD(y))$.

Notice that for $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, since $P(D(u)P(v))$ (resp.
$P(P(u)D(v))$) has the largest $P$-degree in $\phi_{6}(u,v)$ (resp.
$\phi_{7}(u,v)$), the leading monomial of $\phi_{6}(u,v)$ (resp.
$\phi_{7}(u,v)$) with respect to $\leq_{\operatorname{PD}}$ is $P(D(u)P(v))$
(resp. $P(P(u)D(v))$).

###  4.1.  Case of nonunital algebras with $\lambda\neq 0$

Assume in this subsection that $\lambda\neq 0$. We first consider nonunital
free integro-differential ${\mathbf{k}}$-algebras over algebras.

According to the definition of integro-differential algebras, define

${\mathrm{\Phi}_{\mathsf{ID}}}^{\prime}:=\left\\{~{}\phi_{2}(x,y),\phi_{3}(x),\phi_{6}(x,y),\phi_{7}(x,y)~{}\right\\}.$

By Example 3.3, Example 3.4, Example 4.2 and Example 4.3,
${\mathrm{\Phi}_{\mathsf{ID}}}^{\prime}$ is not $\Omega$-GS in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(X)$ with respect to
$\leq_{\operatorname{PD}}$.

######  Example 4.2.

For $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, let

$\begin{array}[]{rcl}f&=&\phi_{7}(u,v)=P(P(u)D(v))-P(u)v+P(uv)+\lambda
P(uD(v)),\\\ \ g&=&\phi_{4}(u,v)=P(u)D(v)-D(P(u)v)+uv+\lambda uD(v),\\\
q&=&P(\star),\\\ p&=&P(P(u)D(v))=\bar{f}=\left.q\right|_{\bar{g}}.\end{array}$

Then

$(f,g)_{p}^{q}=f-\left.q\right|_{g}\equiv-P(D(P(u)v))+P(u)v.$

Let

$\phi_{8}(x,y)=P(D(P(x)y))-P(x)y.$

It is clear that the leading monomial of $\phi_{8}(u,v)$ is $P(D(P(u)v))$ with
respect to $\leq_{\operatorname{PD}}$ which cannot be reduced further.

######  Example 4.3.

For $u,v\in{\mathfrak{S}_{\Omega}}(Z)$, let

$\begin{array}[]{rcl}f&=&\phi_{6}(u,v)=P(D(u)P(v))-uP(v)+P(uv)+\lambda
P(D(u)v),\\\ \ g&=&\phi_{5}(u,v)=D(u)P(v)-D(uP(v))+uv+\lambda D(u)v,\\\
q&=&P(\star),\\\ p&=&P(D(u)P(v))=\bar{f}=\left.q\right|_{\bar{g}}.\end{array}$

Then

$(f,g)_{p}^{q}=f-\left.q\right|_{g}\equiv-P(D(uP(v)))+uP(v).$

Let

$\phi_{9}(x,y)=P(D(xP(y)))-xP(y).$

It is clear that the leading monomial of $\phi_{9}(u,v)$ is $P(D(uP(v)))$ with
respect to $\leq_{\operatorname{PD}}$ which cannot be reduced further.

######  Remark 4.4.

Note that the OPI $\phi_{1}(x,y)$ can be induced by $\phi_{3}(x,y)$ and
$\phi_{6}(x,y)$. So an integro-differential algebra can be seen as a
differential Rota-Baxter algebra. Explicitly, for
$u,v\in{\mathfrak{S}_{\Omega}}(Z)$, let

$\begin{array}[]{rcl}f&=&\phi_{6}(P(u),v)=P(D(P(u))P(v))-P(u)P(v)+P(P(u)v)+\lambda
P(D(P(u))v),\\\ \ g&=&\phi_{3}(u)=D(P(u))-u,\\\ q&=&P(\star P(v)),\\\
p&=&P(D(P(u))P(v))=\bar{f}=\left.q\right|_{\bar{g}}.\end{array}$

Then

$(f,g)_{p}^{q}=f-\left.q\right|_{g}\equiv P(u)P(v)-P(uP(v))-P(P(u)v)-\lambda
P(uv)=\phi_{1}(u,v).$

Now denote ${\mathrm{\Phi}_{\mathsf{ID}}}$ to be the set of the following
OPIs:

  * (1)

$\phi_{1}(x,y)=P(x)P(y)-P(xP(y))-P(P(x)y)-\lambda P(xv)$,

  * (2)

$\phi_{2}(x,y)=D(x)D(y)+\lambda^{-1}D(x)y+\lambda^{-1}xD(y)-\lambda^{-1}D(xy)$,

  * (3)

$\phi_{3}(x)=D(P(x))-x$,

  * (4)

$\phi_{4}(x,y)=P(x)D(y)-D(P(x)y)+xy+\lambda xD(y)$,

  * (5)

$\phi_{5}(x,y)=D(x)P(y)-D(xP(y))+xy+\lambda D(x)y$,

  * (8)

$\phi_{8}(x,y)=P(D(P(x)y))-P(x)y$,

  * (9)

$\phi_{9}(x,y)=P(D(xP(y)))-xP(y)$.

Notice that
${\mathrm{\Phi}_{\mathsf{ID}}}={\mathrm{\Phi}_{\mathsf{DRB}}}\cup\\{\phi_{8}(x,y),\phi_{9}(x,y)\\}$.

######  Proposition 4.5.

$\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}}^{\prime}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}=\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$
for each set $Z$.

###### Proof.

We firstly prove $\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}\subseteq\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}}^{\prime}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$,
which follows from

$\left\\{\begin{array}[]{lll}\phi_{1}(u,v)\in\left\langle\phi_{3}(u,v),\phi_{6}(u,v)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}\
\mathrm{by\ Remark}~{}\ref{phi1},\\\
\phi_{4}(u,v)\in\left\langle\phi_{2}(u,v),\phi_{3}(u)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}\
\mathrm{by\ Example}~{}\ref{phi_4},\\\
\phi_{5}(u,v)\in\left\langle\phi_{2}(u,v),\phi_{3}(u)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}\
\mathrm{by\ Example}~{}\ref{phi_5},\\\
\phi_{8}(u,v)\in\left\langle\phi_{4}(u,v),\phi_{7}(u,v)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}\
\mathrm{by\ Example}~{}\ref{phi8},\\\
\phi_{9}(u,v)\in\left\langle\phi_{5}(u,v),\phi_{6}(u,v)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}\
\mathrm{by\ Example}~{}\ref{phi9},\end{array}\right.$

where $u,v\in{\mathfrak{S}_{\Omega}}(Z)$.

Next we show $\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}}^{\prime}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}\subseteq\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$.
Note that

$\displaystyle P(\phi_{4}(u,v))$
$\displaystyle=P(P(u)D(v))-P(D(P(u)v))+P(uv)+\lambda P(uD(v))$
$\displaystyle=P(P(u)D(v))-P(u)v+P(uv)+\lambda uD(v)-P(D(P(u)v))+P(u)v$
$\displaystyle=\phi_{7}(u,v)-\phi_{8}(u,v),$

and

$\displaystyle P(\phi_{5}(u,v))$
$\displaystyle=P(D(u)P(v))-P(D(uP(v)))+P(uv)+\lambda P(D(u)v)$
$\displaystyle=P(D(u)P(v))-uP(v)+P(uv)+\lambda D(u)v-P(D(uP(v)))+uP(v)$
$\displaystyle=\phi_{6}(u,v)-\phi_{9}(u,v).$

So we have

$\left\\{\begin{array}[]{lll}\phi_{6}(u,v)\in\left\langle\phi_{5}(u,v),\phi_{9}(u,v)\right\rangle_{\Omega\text{-}\mathfrak{Alg}},\\\
\phi_{7}(u,v)\in\left\langle\phi_{4}(u,v),\phi_{8}(u,v)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}.\end{array}\right.$

It proves $\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}}^{\prime}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}\subseteq\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$.

We are done. ∎

Now we can prove ${\mathrm{\Phi}_{\mathsf{ID}}}$ is $\Omega$-GS.

######  Theorem 4.6.

${\mathrm{\Phi}_{\mathsf{ID}}}$ is $\Omega$-GS in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

###### Proof.

Since the ambiguities $i\wedge j$ with $i,j=1,2,3,4,5$ in
${\mathrm{\Phi}_{\mathsf{ID}}}$ are the same as in Theorem 3.7, we only need
to consider the ambiguities involving $\phi_{8}$ and $\phi_{9}$. The cases
that cannot be dealt with directly by Lemma 3.5 are listed below: for
arbitrary
$u,v,w\in{\mathfrak{S}_{\Omega}}(Z),q\in{\mathfrak{S}_{\Omega}}^{\star}(Z)$
and $s\in{\mathfrak{S}_{\Omega}}(Z)\text{ or }\emptyset$,

  * $1\wedge 8$

$P\left(D\left(P\left(u\right)v\right)\right)P\left(w\right)$,
$P\left(u\right)P\left(D\left(P\left(v\right)w\right)\right)$,

  * $3\wedge 8$

$D\left(P\left(D\left(P\left(u\right)v\right)\right)\right)$,

  * $4\wedge 8$

$P\big{(}D\left(P(u)v\right)\big{)}D\left(w\right)$,

  * $5\wedge 8$

$D\left(u\right)P\left(D\left(P\left(v\right)w\right)\right)$,

  * $1\wedge 9$

$P\big{(}D\left(uP(v)\right)\big{)}P\left(w\right)$,
$P\left(u\right)P\big{(}D\left(vP(w)\right)\big{)}$,

  * $3\wedge 9$

$D\Big{(}P\big{(}D\left(uP(v)\right)\big{)}\Big{)}$,

  * $4\wedge 9$

$P\big{(}D\left(uP(v)\right)\big{)}D\left(w\right)$,

  * $5\wedge 9$

$D\left(u\right)P\left(D\left(vP\left(w\right)\right)\right)$,

  * $8\wedge 1$

$P\big{(}D\left(P(u)P(v)s\right)\big{)}$,

  * $8\wedge 4$

$P\big{(}D\left(P(u)D(v)s\right)\big{)}$,

  * $9\wedge 1$

$P\big{(}D\left(sP(u)P(v)\right)\big{)}$,

  * $9\wedge 5$

$P\big{(}D\left(sD(u)P(v)\right)\big{)}$,

  * $8\wedge 8$

$P\left(D\Big{(}P\big{(}D\left(P(u)v\right)\big{)}w\Big{)}\right)$,

  * $8\wedge 9$

$P\left(D\Big{(}P\big{(}D\left(uP(v)\right)\big{)}w\Big{)}\right)$,

  * $9\wedge 8$

$P\left(D\Big{(}uP\big{(}D\left(P(v)w\right)\big{)}\Big{)}\right)$,

  * $9\wedge 9$

$P\left(D\Big{(}uP\big{(}D\left(vP(w)\big{)}\right)\Big{)}\right)$.

All these compositions can be treated similarly as in the proof of Theorem
3.7. We only give the complete proof for the case $8\wedge 1$. Take
$f=\phi_{8}(u,P(v)s)$, $g=\phi_{1}(u,v)$, $p=P(D(P(u)P(v)s))$ and $q=P(D(\star
s))$. Then we have

$\displaystyle(f,g)_{p}^{q}$
$\displaystyle=-P(u)P(v)s+P(D(P(uP(v))s))+P(D(P(P(u)v)s))+\lambda
P(D(P(uv)s))$ $\displaystyle\equiv-P(uP(v))s-P(P(u)v)s-\lambda
P(uv)s+P(uP(v))s+P(P(u)v)s+\lambda P(uv)s$ $\displaystyle\equiv
0\bmod\left(S_{{\mathrm{\Phi}_{\mathsf{ID}}}}(Z),p\right).$

We are done. ∎

######  Theorem 4.7.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ a nonunital
${\mathbf{k}}$-algebra. Then we have:

${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{ID}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)={\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)/\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}.$

Moreover, assume $I_{A}$ has a GS basis $G$ with respect to the degree-
lexicographical order $\leq_{\rm{dlex}}$. Then
$S_{{\mathrm{\Phi}_{\mathsf{ID}}}}(Z)\cup G$ is an operated GS basis of
$\left\langle S_{{\mathrm{\Phi}_{\mathsf{ID}}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$ in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

###### Proof.

Since the leading monomial in ${\mathrm{\Phi}_{\mathsf{ID}}}$ has no subword
in ${\mathcal{S}}(X)\backslash X$, the result follows immediately from Theorem
4.6 and Theorem 2.13. ∎

As a consequence, we obtain a linear basis .

######  Theorem 4.8.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ a nonunital
${\mathbf{k}}$-algebra with a GS basis $G$ with respect to $\leq_{\rm{dlex}}$.
Then a linear basis of the free nonunital $\lambda$-integro-differential
algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{DRB}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
over $A$ is given by the set ${\rm
Irr}(S_{{\mathrm{\Phi}_{\mathsf{ID}}}}(Z)\cup G)$, which is by definition the
complement in ${\mathfrak{S}_{\Omega}}(Z)$ of the subset consisting of
$q|_{w}$ where $w$ runs through

$\bar{s},P(u)P(v),D(u)D(v),D(P(u)),P(u)D(v),D(u)P(v),P(D(P(u)v)),P(D(uP(v)))$

for arbitrary $s\in
G,q\in{\mathfrak{S}_{\Omega}^{\star}}(Z),u,v\in{\mathfrak{S}_{\Omega}}(Z).$

###### Proof.

It can be induced directly from Theorem 2.8. ∎

######  Remark 4.9.

Since the monomial order $\leq_{\operatorname{PD}}$ is different from that
used in [7], our operated GS basis and linear basis are different from theirs.
The reason is that the monomial order in [7] does not satisfy the condition of
Theorem 2.13, thus cannot enable us to discuss free integro-differential
algebras over algebras.

######  Remark 4.10.

Define a new OPI $\phi_{10}(x)=P(D(x))$, and let

$\Phi_{\mathsf{IID}}=\\{~{}\phi_{1}(x,y),\phi_{2}(x,y),\phi_{3}(x),\phi_{10}(x)~{}\\}.$

A $\Phi_{\mathsf{IID}}$-algebra is just a nonunital $\lambda$-integro-
differential algebra with the operators $P$ and $D$ being the inverse operator
of each other, so we call such an operated algebra an invertible integro-
differential algebra. One can show that
$\Phi_{\mathsf{IID}}\cup\\{\phi_{4}(x,y),\phi_{5}(x,y)\\}$ is $\Omega$-GS in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

###  4.2.  Case of nonunital algebras with $\lambda=0$

Now we consider nonunital free integro-differential algebras on algebras with
$\lambda=0$. This case can be studied similarly as the case $\lambda\neq 0$,
so we omit the details in this subsection.

As in Subsection 3.2, for a OPI $\phi$, we denote $\phi^{0}$ for $\phi$ with
$\lambda=0$ and also write $\phi^{0}=\phi$ when $\lambda$ does not appear in
$\phi$ for convenience. Let

${\mathrm{\Phi}_{\mathsf{ID}}^{0}}^{\prime}:=\left\\{~{}\phi_{2}^{0}(x,y),\phi_{3}^{0}(x),\phi_{6}^{0}(x,y),\phi_{7}^{0}(x,y)\right\\}.$

Again, ${\mathrm{\Phi}_{\mathsf{ID}}^{0}}^{\prime}$ is not $\Omega$-GS in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

######  Remark 4.11.

By Example 4.2, we can get $\phi_{8}^{0}(u,v)$ from $\phi_{4}^{0}(u,v)$ and
$\phi_{7}^{0}(u,v)$.

One can not obtain $\phi_{9}^{0}(u,v)$ from
$S_{{\mathrm{\Phi}_{\mathsf{ID}}^{0}}^{\prime}}(Z)$ as in Example 4.3, since
$\phi_{5}$ does not belong to ${\mathrm{\Phi}_{\mathsf{ID}}^{0}}^{\prime}$.
However, we can still generate $\phi_{9}^{0}(u,v)$ as follows: for
$u,v\in{\mathfrak{S}_{\Omega}}(Z)$, let

$\begin{array}[]{rcl}f&=&\phi_{6}^{0}(u,v)=P(D(u)P(v))-uP(v)+P(uv),\\\ \
g&=&\phi_{2}^{0}(u,P(v))=D(u)P(v)+uD(P(v))-D(uP(v)),\\\ q&=&P(\star),\\\
w&=&P(D(u)P(v))=\bar{f}=\left.q\right|_{\bar{g}}.\end{array}$

Then

$(f,g)_{w}=f-\left.q\right|_{g}\equiv P(D(uP(v)))-uP(v)=\phi_{9}^{0}(u,v).$

Now denote ${\mathrm{\Phi}_{\mathsf{ID}}^{0}}$ to be the set of the following
OPIs:

  * (1)

$\phi_{1}^{0}(x,y)=P(x)P(y)-P(xP(y))-P(P(x)y)$,

  * (2)

$\phi_{2}^{0}(x,y)=D(x)y+xD(y)-D(xy)$,

  * (3)

$\phi_{3}^{0}(x)=D(P(x))-x$,

  * (4)

$\phi_{4}^{0}(x,y)=P(x)D(y)-D(P(x)y)+xy$,

  * (8)

$\phi_{8}^{0}(x,y)=P(D(P(x)y))-P(x)y$,

  * (9)

$\phi_{9}^{0}(x,y)=P(D(xP(y)))-xP(y)$.

As in the previous subsection, one can prove the following results.

######  Proposition 4.12.

$\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}^{0}}^{\prime}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}=\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}^{0}}}(Z)\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$
for any set $Z$.

######  Theorem 4.13.

${\mathrm{\Phi}_{\mathsf{ID}}^{0}}$ is $\Omega$-GS in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

######  Theorem 4.14.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ a nonunital
${\mathbf{k}}$-algebra. Then we have:

${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{ID}}^{0}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)={\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)/\left\langle
S_{{\mathrm{\Phi}_{\mathsf{ID}}^{0}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}.$

Moreover, assume $I_{A}$ has a GS basis $G$ with respect to the degree-
lexicographical order $\leq_{\rm{dlex}}$. Then
$S_{{\mathrm{\Phi}_{\mathsf{ID}}^{0}}}(Z)\cup G$ is an operated GS basis of
$\left\langle S_{{\mathrm{\Phi}_{\mathsf{ID}}^{0}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{Alg}}$ in
${\mathbf{k}}{\mathfrak{S}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{PD}}$.

######  Theorem 4.15.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{S}}(Z)/I_{A}$ a nonunital
${\mathbf{k}}$-algebra with a GS basis $G$ with respect to $\leq_{\rm{dlex}}$.
Then the set ${\rm Irr}(S_{{\mathrm{\Phi}_{\mathsf{ID}}^{0}}}(Z)\cup G)$ which
is by definition the complement of

$\left\\{q|_{\bar{s}},q|_{P(u)P(v)},q|_{D(u)v},q|_{D(P(u))},q|_{P(u)D(v)},q|_{P(D(P(u)v))},q|_{P(D(uP(v)))},s\in
G,q\in{\mathfrak{S}_{\Omega}^{\star}}(Z),u,v\in{\mathfrak{S}_{\Omega}}(Z)\right\\}$

in ${\mathfrak{S}_{\Omega}}(Z)$ is a linear basis of the free nonunital
$0$-integro-differential algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{ID}}^{0}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
over $A$.

###  4.3.  Case of unital algebras

Now we consider unital integro-differential algebras. Since the proofs are
similar to those in the previous subsections, we omit most of them. The study
still is divided into cases of $\lambda\neq 0$ and $\lambda=0$.

When $\lambda\neq 0$, since unital integro-differential algebras have the
condition $D(1)=0$, we put
${\mathrm{\Phi}_{\mathsf{uID}}}:={\mathrm{\Phi}_{\mathsf{ID}}}\cup\\{D(1)\\}$.

######  Theorem 4.16.

${\mathrm{\Phi}_{\mathsf{uID}}}$ is $\Omega$-GS in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{uPD}}$.

######  Theorem 4.17.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ a unital
${\mathbf{k}}$-algebra. Then we have:

${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{uID}}}\text{-}{\mathfrak{uAlg}}}_{{\mathfrak{uAlg}}}(A)={\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)/\left\langle
S_{{\mathrm{\Phi}_{\mathsf{uID}}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}.$

Moreover, assume $I_{A}$ has a GS basis $G$ with respect to the degree-
lexicographical order $\leq_{\rm{dlex}}$. Then
$S_{{\mathrm{\Phi}_{\mathsf{uID}}}}(Z)\cup G$ is an operated GS basis of
$\left\langle S_{{\mathrm{\Phi}_{\mathsf{uID}}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}$ in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{uPD}}$.

######  Theorem 4.18.

Let $Z$ be a set, $A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ a unital
${\mathbf{k}}$-algebra with a GS basis $G$ with respect to $\leq_{\rm{dlex}}$.
Then a linear basis of the free unital $\lambda$-integro-differential algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{uID}}}\text{-}{\mathfrak{uAlg}}}_{{\mathfrak{uAlg}}}(A)$
over $A$ is given by the set ${\rm
Irr}(S_{{\mathrm{\Phi}_{\mathsf{uID}}}}(Z)\cup G)$, which is by definition the
complement in ${\mathfrak{M}_{\Omega}}(Z)$ of the subset consisting of
$q|_{w}$ where $w$ runs through

$\bar{s},P(u)P(v),D(u)D(v),D(P(u)),P(u)D(v),D(u)P(v),P(D(P(u)v)),P(D(uP(v))),D(1)$

for arbitrary $s\in
G,q\in{\mathfrak{M}_{\Omega}^{\star}}(Z),u,v\in{\mathfrak{M}_{\Omega}}(Z).$

When $\lambda=0$, denote
${\mathrm{\Phi}_{\mathsf{uID}}^{0}}:={\mathrm{\Phi}_{\mathsf{ID}}^{0}}$.

######  Theorem 4.19.

${\mathrm{\Phi}_{\mathsf{uID}}^{0}}$ is $\Omega$-GS in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{uPD}}$.

######  Theorem 4.20.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ a unital
${\mathbf{k}}$-algebra. Then we have:

${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{uID}}^{0}}\text{-}{\mathfrak{uAlg}}}_{{\mathfrak{uAlg}}}(A)={\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)/\left\langle
S_{{\mathrm{\Phi}_{\mathsf{uID}}^{0}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}.$

Moreover, assume $I_{A}$ has a GS basis $G$ with respect to the degree-
lexicographical order $\leq_{\rm{dlex}}$. Then
$S_{{\mathrm{\Phi}_{\mathsf{uID}}^{0}}}(Z)\cup G$ is an operated GS basis of
$\left\langle S_{{\mathrm{\Phi}_{\mathsf{uID}}^{0}}}(Z)\cup
I_{A}\right\rangle_{\Omega\text{-}\mathfrak{uAlg}}$ in
${\mathbf{k}}{\mathfrak{M}_{\Omega}}(Z)$ with respect to
$\leq_{\operatorname{uPD}}$.

######  Theorem 4.21.

Let $Z$ be a set and $A={\mathbf{k}}{\mathcal{M}}(Z)/I_{A}$ a unital
${\mathbf{k}}$-algebra with a GS basis $G$ with respect to $\leq_{\rm{dlex}}$.
Then the set ${\rm Irr}(S_{{\mathrm{\Phi}_{\mathsf{uID}}^{0}}}(Z)\cup G)$
which is by definition the complement of

$\left\\{q|_{\bar{s}},q|_{P(u)P(v)},q|_{D(u)v},q|_{D(P(u))},q|_{P(u)D(v)},q|_{P(D(P(u)v))},q|_{P(D(uP(v)))},s\in
G,q\in{\mathfrak{M}_{\Omega}^{\star}}(Z),u,v\in{\mathfrak{M}_{\Omega}}(Z)\right\\}$

in ${\mathfrak{M}_{\Omega}}(Z)$ is a linear basis of the free unital
$0$-integro-differential algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{uID}}^{0}}\text{-}{\mathfrak{uAlg}}}_{{\mathfrak{uAlg}}}(A)$
over $A$.

###  4.4. Differential Rota-Baxter algebras vs integro-differential algebras

Since integro-differential algebras have one more defining relation than
differential Rota-Baxter algebras, by Proposition 2.10, the free integro-
differential algebra over an algebra $A$ is a quotient of the free
differential Rota-Baxter algebra over $A$ in general. However, by using the
descriptions of ${\mathrm{\Phi}_{\mathsf{DRB}}}$ and
${\mathrm{\Phi}_{\mathsf{ID}}}$ and Theorems 3.7 and 4.6, we could also show
that the former one is a differential Rota-Baxter algebra subalgebra of the
later one.

######  Theorem 4.22.

The free nonunital $\lambda$-integro-differential algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{ID}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
over an algebra $A$ is differential Rota-Baxter subalgebra of the free
nonunital $\lambda$-differential Rota-Baxter algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{DRB}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
over $A$.

###### Proof.

We have the observation mentioned before

${\mathrm{\Phi}_{\mathsf{ID}}}={\mathrm{\Phi}_{\mathsf{DRB}}}\cup\\{\phi_{8}(x,y),\phi_{9}(x,y)\\}.$

That is to say, the operated Gröbner-Shirshov basis of the free nonunital
$\lambda$-differential Rota-Baxter algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{DRB}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
over an algebra $A$ is a subset of that of the free nonunital
$\lambda$-integro-differential algebra
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{ID}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
over $A$. So by Diamond Lemma,
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{ID}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
is a subspace of
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{DRB}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$.
It is obvious that
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{ID}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$
is also differential Rota-Baxter subalgebra of
${\mathcal{F}}^{{\mathrm{\Phi}_{\mathsf{DRB}}}\text{-}{\mathfrak{Alg}}}_{{\mathfrak{Alg}}}(A)$.
∎

######  Remark 4.23.

Gao and Guo [7] also studied GS bases of the free integro-differential
algebras and free differential Rota-Baxter algebra both generated by sets, and
they deduced that the free integro-differential algebra generated by a set is
a subalgebra of the free differential Rota-Baxter algebra generated by the
same set. Theorem 4.22 proves an analogous fact for these free algebras
generated by algebras. However, our method is completely different from
theirs.

######  Remark 4.24.

By using the descriptions of ${\mathrm{\Phi}_{\mathsf{DRB}}^{0}}$ and
${\mathrm{\Phi}_{\mathsf{ID}}^{0}}$ (resp. ${\mathrm{\Phi}_{\mathsf{uDRB}}}$
and ${\mathrm{\Phi}_{\mathsf{uID}}}$, ${\mathrm{\Phi}_{\mathsf{uDRB}}^{0}}$
and ${\mathrm{\Phi}_{\mathsf{uID}}^{0}}$) and Theorems 3.14 and 4.13 (resp.
Theorems 3.18 and 4.16, Theorems 3.22 and 4.19), we always have the same
result in both unital and nonunital cases with any $\lambda$ (zero or
nonzero).

Acknowledgements: The authors were supported by NSFC (No. 11771085, 12071137)
and by STCSM (22DZ2229014).

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