# Decomposition of Płonka sums
into direct systems

Krystyna Mruczek-Nasieniewska, Mateusz Klonowski

###### Abstract

The Płonka sum is an algebra determined using a structure called direct
system. By a direct system, we mean an indexed family of algebras with
disjoint universes whose indexes form a join-semilattice s.t. if two indexes
are in a partial order relation, then there is a homomorphism from the algebra
of the first index to the algebra of the second index. The sum of the sets of
the direct system determines the universe of Płonka sum. Therefore, to speak
about a Płonka sum, there must be a direct system on which this algebra is
based.

Up to now, Płonka sums have been analyzed only by referring to their direct
systems. In our paper, we will decompose the Płonka sum without knowing any of
direct systems that determine it. The decomposition we describe will be based
on the method of determining all direct systems of a given Płonka sum. Our
analysis also allows us to determine, in the case of any algebra that meets
the assumptions made in the article, whether it is a Płonka sum. The proposed
method is based on two concepts introduced in the paper: isolated algebra and
Płonka homomorphism.

Key words: decomposition, direct frame, direct system, isolated algebra,
Płonka homomorphism, Płonka sums

MSC: 06F99, 08A99

## Introduction

In [18] J. Płonka defined a certain algebraic structure using the so-called
direct system. This structure is called Płonka sum. By a direct system, we
mean an indexed family of algebras with disjoint universes whose indexes form
a join-semilattice s.t. if two indexes are in a partial order relation, then
there is a homomorphism from the algebra of the first index to the algebra of
the second index. The sum of the sets of the direct system determines the
universe of Płonka sum. Therefore, to speak about a Płonka sum, there must be
a direct system on which this algebra is based.

Płonka sum was constructed to analyze equational classes determined by certain
regular identities, i.e., such identities in the case of which the sets of
variables of the left and right sides are equal. Considering classes defined
by regular identities is part of research on identities, in which the
structure of terms creating them matters. Example of such identities are:
normal, externally compatible, P-compatible, biregular, symmetric and many
others (see for example [5, 11, 12, 14, 15, 16, 18, 19, 22, 24]). In recent
years, Płonka sums have been analyzed not only within universal algebra but
also logic. For example, they are used as part of semantic analysis of
variable inclusion logic (see [1, 2, 17]). Such logic is an example of a
formal system whose semantics take into account not only logical connections
but also other dependencies, and relations between sentences (cf. [4, 6, 7,
8]).

Note that given a direct system, we can consider the Płonka sum based on it.
However, it is not obvious how we could recreate this one or perhaps another
direct system of this Płonka sum. Therefore, we can analyze Płonka sum from
two perspectives. From the first one, the direct system of a given Płonka sum
is determined and known. In other words, the first perspective employs the so-
called direct system to determine the Płonka sum. From the second one, the
Płonka sum is presented as a universe on which operations are defined. Thus,
the second perspective acknowledges the Płonka sum as an algebra with a
universe that can be expressed as a union of a family of sets forming a
certain direct system. Of course, a direct system for a given Płonka sum must
exist in both cases. The difference is that in the first case, it is known and
in the second it is not. So far, the first perspective has been adopted in the
literature without asking how, given the algebra being the Płonka sum, all the
direct systems that determine it can be identified.

In our paper, we will present a method for determining direct systems of a
given Płonka sum by decomposing this sum so as to obtain an appropriate
indexed family of algebras with disjoint universes forming semilattice and an
appropriate set of homomorphisms. In other words, we will represent direct
systems by structures we can receive by our method of Płonka sums
decomposition. Such an approach fits into the investigations on structures’
representation. The well-known example of a result related to this topic can
be Stone’s representation theorem for Boolean algebras. Moreover, since the
universe of the Płonka sum is a set-theoretic union, it seems natural to ask
about the decomposition of the Płonka sum, which takes into account the
division of its universe into disjoint sets. Such a problem fits into the
general investigations on complex structures’ decomposition (or
factorization). The well known example of a result related to this topic is
the fundamental theorem of arithmetic or the fundamental subdirect
representation theorem of G. Birkhoff.

There are some constructions known in the literature that propose some
decompositions of Płonka sums (see [9, 21]). The first of the decompositions
was presented in [9]. It indicates (with certain assumptions) subdirectly
irreducible algebras in the class of all Płonka sums. It has been proven that
they are either subdirect irreducible algebras in the initial class or are
created from them after adding the so-called absorbing element. The second of
the decompositions was presented in [21]. This work shows that each Płonka sum
can be represented as a subdirect product of two Płonka sums obtained from the
initial sum, the so-called sum with zero and sum with unit.

However, both known decompositions are made assuming we know the direct system
determining the initial Płonka sum. As we mentioned, in our paper, we will
decompose algebras being Płonka sums without knowing their direct systems. Our
analysis also allows determine in the case of any algebra that meets the
assumptions adopted in the article whether it is a Płonka sum.

The article consists of five sections. In Section 1, we present some
preliminary notions used in the paper. In Section 2, we introduce the
definition of Płonka sum. In Section 3, we introduce key concepts used in our
decomposition method: isolated algebras and Płonka homomorphism. In Section 4,
we present the main result of the paper — the decomposition theorem. Section
5, contains some exemplary illustrations of the decomposition method we
propose. Appendix contains examples of algebra characterized by some
definitions we present. In some cases, the algebras discussed in Appendix
allow us to better formulate various problems that interest us in this
article.

##  1 Preliminaries

Let $\tau\colon F\rightarrow N$ be a type of algebras where $F$ is a set of
fundamental operation symbols and $N$ is the set of all positive integers. For
an algebra $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$, by $ar(k)$
(where $k\leq n$) we denote the arity of operation $F_{k}^{\boldsymbol{A}}$.
We will denote the restriction of the function $F_{k}^{\boldsymbol{A}}$ to the
set $B^{ar(k)}\subseteq A^{ar(k)}$ by
$F_{k}^{\boldsymbol{A}}\\!\\!\upharpoonright_{B}$. We will say that two
algebras are of the same type (similar) if they have the same number of
operations with the same arity. In the sequel of the paper, we will analyze
algebras having at least one operation of arity not smaller than $2$. As we
will find out, Płonka sums of algebras in which all operations are unary are
not interesting.

One of the main concepts used in the paper is the concept of join-semilattice.
We will sometimes consider this structure as a pair $\langle A,R\rangle$,
where $R$ is a partial order on $A$ and for any $x,y\in A$, there is the least
upper bound of the set $\\{x,y\\}$, and sometimes as a structure with one two-
argument the least upper bound operation $\vee$.

Let $\boldsymbol{A}$ and $\boldsymbol{B}$ be similar algebras. By
$\mathsf{dom}(\boldsymbol{A})$ we will denote the universe of
$\boldsymbol{A}$, and by $\mathsf{Hom}(\boldsymbol{A},\boldsymbol{B})$ we
denote the set of all homomorphisms from $\boldsymbol{A}$ to $\boldsymbol{B}$.

Let $\boldsymbol{B},\boldsymbol{C}$ be subalgebras of $\boldsymbol{A}$. For
any $\varphi\in\mathsf{Hom}(\boldsymbol{B},\boldsymbol{C})$, we define
function $f_{\varphi}\colon A\longrightarrow A$ in the following way:

$f_{\varphi}(a)=\begin{cases}\varphi(a),\quad&\text{ if }a\in B\\\
a,\quad&\text{ if }a\notin B.\end{cases}$

By $\mathsf{max}$ and $\mathsf{min}$, we will denote the greatest element and
the least element of a given set, respectively, with respect to a given order
if such elements exist.

##  2 Płonka Sum

We present now Płonka sum formally. We start with a definition of a direct
system.

######  Definition 1.

A direct system is a triple
$\mathcal{A}=\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in
I},\allowbreak\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$ s.t.:

  * a.

$\mathcal{I}=\langle I,\leqslant\rangle$ is a join-semilattice,

  * b.

$\\{\boldsymbol{A}_{i}\\}_{i\in I}$ a non-empty family of algebras of the same
type and with disjoint universes,

  * c.

$\\{\varphi_{i,j}\\}_{i,j\in I}$ is a family of homomorphism s.t. for all
$i,j\in I$:

    * –

if $i\leqslant j$, then there is a homomorphism s.t.
$\varphi_{i,j}:A_{i}\longrightarrow A_{j}$,

    * –

if $i\leqslant j\leqslant k$, then
$\varphi_{j,k}\circ\varphi_{i,j}=\varphi_{i,k}$,

    * –

$\varphi_{i,i}=\textsf{id}_{A_{i}}$.111For simplicity of notation, we will
omit identity homomorphism in our considerations — we will assume by default
that $\varphi_{i,i}=\textsf{id}_{A_{i}}$.

Additionally, we adopt the following notions:

  * •

a direct system $\mathcal{A}$ is called a non-trivial one iff there are
$i,j\in I$ s.t. $i\neq j$,

  * •

an ordered pair $\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$
s.t. the conditions a and b are satisfied we call a direct frame.

Let us now introduce the definition of Płonka sum.

######  Definition 2.

Let $\mathcal{A}=\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in
I},\allowbreak\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$ be a direct
system. A Płonka sum (in symb.: $S(\mathcal{A})$) is an algebra $\langle
A;F_{1}^{\boldsymbol{A}},\ldots F_{n}^{\boldsymbol{A}}\rangle$ similar to
algebras of family $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ s.t.:

  * •

$A=\bigcup_{i\in I}A_{i}$,

  * •

for any $k\leq n$, for all $x_{1},\ldots,x_{ar(k)}\in A$:

$F_{k}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(k)}):=F_{k}^{\boldsymbol{A}_{m}}(\varphi_{i_{1},m}(a_{1}),\ldots,\varphi_{i_{ar(k)},m}(a_{ar(k)})),$

where for any $j\leq ar(k)$, $a_{j}\in A_{i_{j}}$ and $m=i_{1}\vee\ldots\vee
i_{ar(k)}$.

In Appendix, we present several examples of Płonka sums, which we will refer
to in the sequel of our paper.

Having introduced the definitions of the direct system and Płonka sum, we can
try to compare our paper’s main goal with the fundamental arithmetic theorem,
Birkhoff’s subdirect representation theorem, and Stone’s representation
theorem for Boolean algebras. Examining the Płonka sum by considering its
direct systems can be compared to considering a given number as the product of
its prime factors. In turn, the question about direct systems that determine a
given Płonka sum can be compared to the problem of determining the prime
factors of a given number. We can also make a comparison with Birkhoff’s
theorem. In the case of decomposing an algebra into subdirect products, the
aim is to obtain algebras that are less complex than the initial algebra and
which, when properly combined, allow us to obtain the initial algebra. Our
paper aims to decompose an algebraic structure into ‘smaller’ algebras so that
we receive the former by union of the latter. On the other hand, Stone’s
theorem can also be recalled as a good reference point for our research. In
our article, we will show that every direct system is isomorphic to a certain
system of isolated algebras — we define the notion system of isolated algebras
in the sequel.

Since our goal is to indicate the direct system for an algebra being the
Płonka sum, we have to determine the following components:

  * $\blacktriangleright$

an indexed family of algebras with disjoint universes where the set of indexes
form a join-semilattice,

  * $\blacktriangleright$

a set of homomorphisms satisfying given conditions.

In the next section, we will reconstruct both the family of algebras and the
set of homomorphisms. Then we will show that the obtained system will in fact
define the initial algebra being Płonka sum, which will mean that the goal of
our article has been achieved.

##  3 Isolated Algebras and Płonka Homomorphism

Let us now introduce the notion of an isolated algebra and a Płonka
homomorphism — the key notions of our decomposition method of Płonka sum.

###  3.1 Isolated Algebras

######  Definition 3.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ and
$\boldsymbol{B}=\langle
B;F_{1}^{\boldsymbol{B}},\ldots,F_{n}^{\boldsymbol{B}}\rangle$ be algebras of
the same type. Algebra $\boldsymbol{B}$ is an _algebra isolated wrt
$\boldsymbol{A}$_ (an isolated algebra) iff the following conditions are
satisfied:

  * •

$\boldsymbol{B}$ is a subalgebra of $\boldsymbol{A}$,

  * •

for every $m\leq n$, for all $a_{1},\dots,a_{ar(m)}\in A$, if there is $i\leq
ar(m)$ s.t. $a_{i}\not\in B$, then
$F_{m}^{\boldsymbol{A}}(a_{1},\dots,a_{ar(k)})\not\in B$.

The examples of algebras isolated wrt a Płonka sum were presented in Example
29 (see Appendix).

As can be easily seen, in the case of algebras of type (1), each of their
subalgebras will be isolated for the obvious reason. As we stated above, we
will focus on algebras that have at least one operation of arity greater than
$1$. Moreover, we will consider algebras without constants.

Additionally, in the article, we will only consider algebras with respect to
which there are finitely many isolated algebras. This does not mean, however,
that the algebras we are considering, in particular the Płonka sums, must be
finite or that algebras isolated wrt them are finite. In Example 30 (see
Appendix), an infinite algebra (Płonka sum) is specified, with respect to
which we can determine finitely many infinite isolated algebras.

Note that in Example 30 (see Appendix), we indicated for the Płonka sum its
direct system consisting of two algebras. The first one, with the smaller
index, is an isolated algebra. The second one, with the greater index, is not
isolated. But the union of the first and the second is an isolated algebra. We
can generalize this observation. Namely, if we consider a union of the
universe of a given algebra and the universes of all algebras with the smaller
indexes, we will obtain an isolated algebra.

Let $\mathcal{A}=\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in
I},\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$ be a direct system. For
every $i\in I$, we put:

$A_{i}^{+}:=\bigcup_{j\leqslant i}A_{j}.$

We can observe that for every $i\in I$, $A_{i}^{+}\neq\emptyset$. Moreover, we
can prove that for every $i\in I$, $A_{i}^{+}$ is closed under operations of
$S(\mathcal{A})$.

######  Fact 4.

Let $\mathcal{A}=\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in
I},\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$ be a direct system.
Then, $A^{+}_{i}$ is closed under operations of $S(\mathcal{A})$.

###### Proof.

Let $i\in I$, $m\leq n$ and $a_{1},\ldots,a_{ar(m)}\in A_{i}^{+}$. By
Definition 2, $F^{S(\mathcal{A})}_{m}(a_{1},\allowbreak\ldots,\allowbreak
a_{ar(m)})=F^{\boldsymbol{A}_{l}}_{m}(\varphi_{i_{1},l}(a_{1}),\ldots,\allowbreak\varphi_{i_{ar(m)},l}(\allowbreak
a_{ar(m)}))$, where $l=i_{1}\vee\ldots\vee i_{ar(m)}$ and for every $j\leq
ar(m)$, $a_{j}\in A_{i_{j}}$. By (3.1) and Definition 1,
$i_{1},\ldots,i_{ar(m)}\leqslant i$. Thus, $l\leqslant i$. Therefore,
$A_{l}\subseteq A_{i}^{+}$, so
$F^{S(\mathcal{A})}_{m}(a_{1},\ldots,a_{ar(m)})\in A_{i}^{+}$. ∎

Note that the following properties hold for any algebra
$\boldsymbol{A}^{+}_{i}$ obtained from the algebra $\boldsymbol{A}_{i}$.

######  Fact 5.

Let $\mathcal{A}=\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in
I},\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$ be a direct system.
Then:

  1. 1.

for all $i\in I$, $\boldsymbol{A}^{+}_{i}$ is an algebra isolated wrt
$S(\mathcal{A})$,

  2. 2.

for all $i,j\in I$, $A_{i}^{+}\subseteq A_{j}^{+}$ iff $i\leqslant j$,

  3. 3.

there is $i\in I$ s.t. $\boldsymbol{A}_{i}^{+}=S(\mathcal{A})$,

  4. 4.

for all $i,j\in I$, $A_{i}\cap A_{j}^{+}\neq\emptyset$ iff $i\leqslant j$,

  5. 5.

for all $i,j\in I$, $A^{+}_{i}=A_{j}^{+}$ iff $i=j$,

  6. 6.

for all $n\geq 2$, if $i_{1},\ldots,i_{n}\in I$ and
$A_{i_{1}}^{+}\cap\ldots\cap A_{i_{n}}^{+}\neq\emptyset$, then there is $i\in
I$ s.t. $A^{+}_{i}=A_{i_{1}}^{+}\cap\ldots\cap A_{i_{n}}^{+}$.

###### Proof.

Ad 1. Let $i\in I$, $m\leq n$ and $a_{1},\ldots,a_{ar(m)}\in\bigcup_{j\in
I}A_{j}$. We assume that for some $l\leq ar(m)$, $a_{l}\notin A_{i}^{+}$.
Thus, for all $j\leqslant i$, $a_{l}\notin A_{j}$. Then, by Definition 3, for
all $j\leqslant i$, $F_{m}^{S(\mathcal{A})}(a_{1},\ldots,a_{ar(m)})\notin
A_{j}$. Therefore, by (3.1), $F_{m}^{S(\mathcal{A})}(a_{1},\ldots
a_{ar(m)})\notin A_{i}^{+}$.

Ad 2. Let $i,j\in I$. ($\rightarrow$) Suppose $A_{i}^{+}\subseteq A_{j}^{+}$.
Thus, by (3.1), $A_{i}\subseteq A_{j}^{+}$, so for any $a\in A_{i}$, there is
$k\leqslant j$ s.t. $a\in A_{k}$. By Definition 1, for all $k\in I$, $i=k$ or
$A_{i}\cap A_{k}=\emptyset$. Therefore, $i\leqslant j$. ($\leftarrow$) By
(3.1), if $i\leqslant j$, then $A_{i}^{+}\subseteq A_{j}^{+}$.

Ad 3. By the assumption concerning the algebras we consider in the article and
1, $\mathcal{I}$ is a finite join-semilattice.

Ad 4. Let $i,j\in I$. ($\rightarrow$) Suppose that $A_{i}\cap
A_{j}^{+}\neq\emptyset$. Let $a\in A_{i}\cap A_{j}^{+}$. Since $a\in
A_{j}^{+}$, by (3.1), $a\in A_{j_{0}}$, where $j_{0}\leqslant j$. Thus,
$A_{i}\cap A_{j_{0}}\neq\emptyset$. By Definition 1, ${i}={j_{0}}\leqslant j$.
($\leftarrow$) By 3, if $i\leqslant j$, then $A_{i}\cap
A_{j}^{+}\neq\emptyset$.

Ad 5. Let $i,j\in I$. ($\rightarrow$) Suppose that $A_{i}^{+}=A_{j}^{+}$.
Thus, by (3.1), $\bigcup_{k\leqslant i}A_{k}=\bigcup_{k\leqslant j}A_{k}$.
Hence, $A_{i}\subseteq A_{j}^{+}$ and $A_{j}\subseteq A_{i}^{+}$. By 4, $i=j$.
Therefore, $A_{i}=A_{j}$. ($\leftarrow$) Obvious. By Definition 1, if $i=j$,
then $A_{i}=A_{j}$, so by (3.1), $A_{i}^{+}=A_{j}^{+}$.

Ad 6. Let $B:=A_{i_{1}}^{+}\cap\ldots\cap A_{i_{n}}^{+}$ and $X:=\\{A_{j}:j\in
I\text{ and }A_{j}\cap B\neq\emptyset\\}$. Note that $X$ is finite and non-
empty. Thus, $X=\\{A_{j_{1}},\ldots,A_{j_{k}}\\}$, for some $k\geq 1$. Let
$l:=j_{1}\vee\ldots\vee j_{k}$. By definition of $X$,
$A_{i_{1}}^{+}\cap\ldots\cap A_{i_{n}}^{+}\subseteq A_{l}^{+}$. By 4, for all
$o\leq k$, for all $p\leq n$ $j_{o}\leqslant i_{p}$. Thus, $A_{l}^{+}\subseteq
A_{i_{1}}^{+}\cap\ldots\cap A_{i_{n}}^{+}$. ∎

Let us now proceed to the analysis of the set of isolated algebras in terms of
order and other set-theoretic properties.

The set of algebras isolated wrt a given algebra can be partially ordered
using the inclusion relation. As the following fact shows, the relation of
being an isolated algebra is also a relation of partial order.

######  Fact 6.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ be a set of algebras isolated wrt
$\boldsymbol{A}$. Then, for all $i,j\in I$, $\boldsymbol{A}_{i}$ is isolated
wrt $\boldsymbol{A}_{j}$ iff $A_{i}\subseteq A_{j}$.

###### Proof.

($\rightarrow$) By Definition 3. ($\leftarrow$) Suppose $A_{i}\subseteq
A_{j}$. Since $\boldsymbol{A}_{i},\boldsymbol{A}_{j}$ are subalgebras of
$\boldsymbol{A}$, $\boldsymbol{A}_{i}$ is a subalgebra of
$\boldsymbol{A}_{j}$. Moreover, for any $m\leq n$, for all
$a_{1},\ldots,a_{ar(m)}\in A_{j}$, there is $l\leq ar(m)$ s.t. $a_{l}\notin
A_{i}$. Then, since $\boldsymbol{A}_{i}$ is isolated wrt $\boldsymbol{A}$ and
$A_{j}\subseteq A$, we have
$F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\notin A_{i}$. ∎

Regarding set-theoretic operations defined on the set of isolated algebras,
let us first note that the set difference of an algebra and the isolated
algebra is either an empty set or some subalgebra of the initial algebra.

######  Fact 7.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ be a set of algebras isolated wrt
$\boldsymbol{A}$. Then, for all $i,j\in I$, if $A_{i}\subsetneq A_{j}$, then
$A_{j}\setminus A_{i}$ is closed under operations of $\boldsymbol{A}$.

###### Proof.

By Definition 3. ∎

For our further considerations, the observation that a finite, non-trivial
union of isolated incomparable algebras is not an algebra will also be
important.

######  Fact 8.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ be a set of algebras isolated wrt
$\boldsymbol{A}$. Then, if there is no $i\in I$ s.t.
$\bigcup_{j\in(I\setminus\\{i\\})}A_{j}\subseteq A_{i}$ (there is no greatest
element in $\\{\boldsymbol{A}_{i}\\}_{i\in I}$), then there is no subalgebra
of $\boldsymbol{A}$ with the universe $\bigcup_{i\in I}A_{i}$.

###### Proof.

We use induction on the cardinality of $I$.

Base step. For $I=\\{i,j\\}$. Let
$a\in\boldsymbol{A}_{i}\setminus\boldsymbol{A}_{j}$ and
$b\in\boldsymbol{A}_{j}\setminus\boldsymbol{A}_{i}$. Let $m\leq n$ and
$c_{1},\ldots,c_{ar(m)}\in\\{a,b\\}$, where $c_{1}=a$ and $c_{2}=b$. Then, by
Definition 3, $F_{m}^{\boldsymbol{A}}(c_{1},\ldots,c_{ar(m)})\not\in A_{i}\cup
A_{j}$.

Inductive step. Let $I=\\{i_{1},\ldots,i_{l+1}\\}$, where $l\geq 2$. Suppose
there is no $i\in I$ s.t. $\bigcup_{j\in(I\setminus\\{i\\})}A_{j}\subseteq
A_{i}$.

Let $J=I\setminus\\{i_{l+1}\\}$. We consider two cases:

  1. 1.

there is no $i\in J$ s.t. $\bigcup_{j\in(J\setminus\\{i\\})}A_{j}\subseteq
A_{i}$,

  2. 2.

there is $i\in J$ s.t. $\bigcup_{j\in(J\setminus\\{i\\})}A_{j}\subseteq
A_{i}$.

Case 1. By the inductive hypothesis there is no algebra with universe
$\bigcup_{i\in J}A_{i}$. Suppose that for $m\leq n$ and
$a_{1},\ldots,a_{ar(m)}\allowbreak\in\bigcup_{i\in J}A_{i}$,
$F_{m}^{\boldsymbol{A}}(a_{1},\allowbreak\ldots,\allowbreak
a_{ar(m)})\notin\bigcup_{i\in J\setminus\\{i_{l+1}\\}}A_{i}$. We consider two
cases:

  1. 1a.

$F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})=a\in A_{i_{l+1}}$,

  2. 1b.

$F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})=a\notin A_{i_{l+1}}$.

Case 1a. Let $b\in\bigcup_{i\in J}A_{i}$ and $b\notin A_{i_{l+1}}$. Let us
consider $c_{1},\ldots,c_{ar(m)}\in\\{a,b\\}$, where $c_{1}=a$ and $c_{2}=b$.
By Definition 3,
$F_{m}^{\boldsymbol{A}}(c_{1},\ldots,c_{ar(m)})\not\in\bigcup_{i\in I}A_{i}$.
Case 1b. Obvious.

Case 2. Suppose for $i_{0}\in J$,
$\bigcup_{j\in(J\setminus\\{i_{0}\\})}A_{j}\subseteq A_{i_{0}}$. Let $a\in
A_{i_{l+1}}$ and $a\notin A_{i_{0}}$. Thus,
$a\notin\bigcup_{j\in(I\setminus\\{i_{l+1}\\})}A_{j}$. Moreover,
$b\in\bigcup_{j\in(J\setminus\\{i_{l+1}\\})}A_{j}$ and $b\not\in A_{i_{l+1}}$.
Let us consider $c_{1},\ldots,c_{ar(m)}\in\\{a,b\\}$, where $c_{1}=a$ and
$c_{2}=b$. By Definition 3,
$F_{m}^{\boldsymbol{A}}(c_{1},\ldots,c_{ar(m)})\not\in\bigcup_{i\in I}A_{i}$.
∎

In turn, the non-empty intersection of a finite number of isolated algebras is
an isolated algebra.

######  Fact 9.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ be a set of algebras isolated wrt
$\boldsymbol{A}$. Then, if $B:=\bigcap_{i\in I}A_{i}\neq\emptyset$, then
$\langle
B,F^{\boldsymbol{A}}_{1}\\!\\!\upharpoonright_{B},\ldots,F^{\boldsymbol{A}}_{n}\\!\\!\upharpoonright_{B}\rangle$
is algebra isolated wrt $\boldsymbol{A}$.

###### Proof.

Suppose $B:=\bigcap_{i\in J}A_{i}\neq\emptyset$. Obviously, $B$ is closed
under operations of $\boldsymbol{A}$. Let $m\leq n$ and
$a_{1},\ldots,a_{ar(m)}\in A$. Let us assume that for $l\leq ar(m)$,
$a_{l}\not\in B$. Thus, there is $i\in I$ s.t. $a_{l}\notin A_{i}$. By
Definition 3, $F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\not\in A_{i}$.
Therefore, $F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\not\in B$ ∎

By Fact 7 and Fact 8 we receive the following corollary:

######  Corollary 10.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ be a set of algebrs isolated wrt
$\boldsymbol{A}$. Then, for all $i\in I$,
$A_{i}\setminus\bigcup\\{A_{j}\subseteq A:j\in I\text{ and }A_{j}\subsetneq
A_{i}\\}\neq\emptyset$ and $A_{i}\setminus\bigcup\\{A_{j}\subseteq A:j\in
I\text{ and }A_{j}\subsetneq A_{i}\\}$ is closed under operations of
$\boldsymbol{A}$.

###### Proof.

Let $i\in I$. Let us consider the maximal elements of $\\{A_{j}\subseteq
A:j\in I\text{ and }A_{j}\subsetneq A_{i}\\}$ and denote them by
$A_{j_{1}},\ldots,A_{j_{k}}$. If $k=1$, then we use Fact 7. If $k\neq 1$,
then, by Fact 8, $A_{i}\setminus(A_{j_{1}}\cup\ldots\cup
A_{j_{k}})\neq\emptyset$ and, by Fact 7,
$A_{i}\setminus(A_{j_{1}}\cup\ldots\cup A_{j_{k}})$ is closed under operations
of $\boldsymbol{A}$. ∎

We can now show that the set of all isolated algebras forms a join-
semilattice.

######  Fact 11.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ be the set of all algebras isolated
wrt $\boldsymbol{A}$. Then, $\langle\\{\boldsymbol{A}_{i}\\}_{i\in
I},\allowbreak\subseteq\rangle$ is a join-semilattice.

###### Proof.

For all $i,j\in I$, we put:
$\mathsf{dom}(\boldsymbol{A}_{i}\vee\boldsymbol{A}_{j})=\bigcap\\{A_{k}\subseteq
A:k\in I\text{ and }A_{i}\cup A_{j}\subseteq A_{k}\\}$ and for every $m\leq
n$,
$F^{\boldsymbol{A}_{i}\vee\boldsymbol{A}_{j}}_{m}:=F^{\boldsymbol{A}}_{m}\\!\upharpoonright_{\mathsf{dom}(\boldsymbol{A}_{i}\vee\boldsymbol{A}_{j})^{ar(m)}}$.
Obviously $\boldsymbol{A}_{i}\vee\boldsymbol{A}_{j}$ is an algebra. By Fact 9,
$\boldsymbol{A}_{i}\vee\boldsymbol{A}_{j}$ is isolated wrt $\boldsymbol{A}$. ∎

Given the family $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ of algebras isolated wrt
$\boldsymbol{A}$ we can define poset on the set of indexes $I$. Namely,
$\mathcal{I}:=\langle I,\leqslant\rangle$, where for any $i,j\in I$:

$i\leqslant j\text{ iff }A_{i}\subseteq A_{j}.$

We assume that $i<j$ iff $i\leqslant j$ and $i\neq j$. Of course, if
$\\{\boldsymbol{A}_{i}\\}_{i\in I}$ is a join-semilattice wrt $\subseteq$,
then $\mathcal{I}$ is also a join-semilattice wrt $\leqslant$.

Let us also note that since we consider finite sets of isolated algebras, the
relation defined as follows will be non-empty:

$i\prec j\text{ iff }i<j\text{ and there is no $k\in I$ s.t. $i<k<j$},$

for any $i,j\in I$.

By Definition 1, the structure
$\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$, where
$\\{\boldsymbol{A}_{i}\\}_{i\in I}$, is the set of all algebras isolated wrt a
given Płonka sum $\boldsymbol{A}$, is of course not a direct frame of
$\boldsymbol{A}$ — family $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ is not composed
of algebras with disjoint universes. However, as we will show in sequel, for
each direct frame $\langle J,\\{\boldsymbol{B}_{j}\\}_{j\in J}\rangle$ of a
given Płonka sum, the structure $\langle J,\\{\boldsymbol{B}_{j}^{+}\\}_{j\in
J}\rangle$ is isomorphic to some substructure of
$\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$. Now we will
define the substructures that interest us.

######  Definition 12.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ be a set of all algebras isolated wrt
$\boldsymbol{A}$. A frame of algebras isolated wrt $\boldsymbol{A}$ is an
ordered pair $\langle\mathcal{J},\\{\boldsymbol{A}_{j}\\}_{j\in J}\rangle$
s.t. the following conditions are satisfied:

  1. F1.

$J\subseteq I$ and $\\{\boldsymbol{A}_{j}\\}_{j\in J}$ is a non-trivial join-
semilattice wrt $\subseteq$,

  2. F2.

$\mathsf{max}I\in J$,

  3. F3.

for any $j\in I$, if there is $m\in\mathbb{N}$ s.t.
$j_{1},\allowbreak\ldots,\allowbreak j_{m}\in J$ and
$A_{j}=A_{j_{1}}\cap\ldots\cap A_{j_{m}}$, then $j\in J$.

The indicated conditions F2 and F3 guarantee that the frame of isolated
algebras has the following properties:

######  Fact 13.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be a Płonka sum
and $\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$ be a frame
of algebras isolated wrt $\boldsymbol{A}$. Then:

  1. 1.

$\bigcup_{i\in I}A_{i}=A$,

  2. 2.

for all $i,j\in I$, if $i\neq j$ and $i,j$ are minimal elements of
$\mathcal{I}$, then $A_{i}\cap A_{j}=\emptyset$

  3. 3.

for every $a\in A$, set $\\{i\in I:a\in A_{i}\\}$ is non-empty and contains
the least element.

###### Proof.

Ad 1. By F2, $\bigcup_{i\in I}A_{i}=A_{\mathsf{max}I}=A$.

Ad 2. Suppose that $i\neq j$ and $i,j$ are minimal elements of $\mathcal{I}$.
Assume indirectly that $A_{i}\cap A_{j}\neq\emptyset$. By Fact 9, there is an
algebra isolated wrt $\boldsymbol{A}$ with the universe $A_{i}\cap A_{j}$.
Assume its index is $k$. By F3, $k\in I$. Then, by the definition of
$\leqslant$, $k<i,j$.

Ad 3. Let $a\in A$. By Fact 9, $\bigcap\\{A_{i}\subseteq A:i\in I\text{ and
}a\in A_{i}\\}$ is an algebra and, by the definition of $\leqslant$ and and
Definition 12, its index is $\mathsf{min}\\{i\in I:a\in A_{i}\\}$. ∎

As we will see, frames of isolated algebras allow us to describe all direct
systems of a given sum. For now, let us note that each frame of algebras
isolated wrt Płonka sum determines this Płonka sum.

######  Fact 14.

Let $\boldsymbol{A}=\langle A;F_{1}^{\boldsymbol{A}},\ldots,\allowbreak
F_{n}^{\boldsymbol{A}}\rangle$ be a Płonka sum and
$\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$ be a frame of
algebras isolated wrt $\boldsymbol{A}$. Let
$\boldsymbol{B}=\langle\bigcup_{i\in
I}A_{i};F_{1}^{\boldsymbol{B}},\ldots,F_{n}^{\boldsymbol{B}}\rangle$ be an
algebra s.t. for all $m\leq n$, for all $a_{1},\ldots,a_{ar(m)}\in A$:

$F^{\boldsymbol{B}}_{m}(a_{1},\ldots,a_{ar(m)}):=F_{m}^{\boldsymbol{A}_{j}}(a_{1},\ldots,a_{ar(m)}),$

where for all $l\leq ar(m)$, $i_{l}=\mathsf{min}\\{i\in I:a_{l}\in A_{i}\\}$
and $j=i_{1}\vee\ldots\vee i_{ar(m)}$. Then:

$\boldsymbol{A}=\boldsymbol{B}.$

We will now show that the algebras obtained according to the definition of
(3.1) from the direct system form an isomorphic structure with some frame of
isolated algebras.

######  Fact 15.

Let $\mathcal{A}=\langle\langle
I,\leqslant\rangle,\\{\boldsymbol{A}_{i}\\}_{i\in
I},\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$ be a direct system and
$\langle\langle J,\sqsubseteq\rangle,\\{\boldsymbol{B}_{j}\\}_{j\in J}\rangle$
be a frame of algebras isolated wrt $S(\mathcal{A})$, where
$\\{\boldsymbol{B}_{j}\\}_{j\in J}$ is the set of all algebrs isolated wrt
$S(\mathcal{A})$. Then:

  1. 1.

$f:I\longrightarrow J$ s.t. $f(i):=j$, where $A_{i}^{+}=B_{j}$, is injection,

  2. 2.

$\langle\langle f(I),\sqsubseteq\rangle,\\{\boldsymbol{B}_{k}\\}_{k\in
f(I)}\rangle$ and $\langle\langle
I,\leqslant\rangle,\\{\boldsymbol{A}_{i}^{+}\\}_{i\in I}\rangle$ are
isomorphic,

  3. 3.

$\langle\langle f(I),\sqsubseteq\rangle,\\{\boldsymbol{B}_{k}\\}_{k\in
f(I)}\rangle$ is a frame of algebras isolated wrt $S(\mathcal{A})$.

###### Proof.

Ad 1. By Fact 5.1 and Definition 1, for any $i\in I$, there is exactly one
$j\in J$ s.t. $A_{i}^{+}=B_{j}$. Due to the disjoint universes of algebras
from $\\{A_{i}\\}_{i\in I}$, $f$ is an injection.

Ad 2. By 1, function $g\colon I\longrightarrow f(I)$ s.t. $g(i):=f(i)$ is a
bijection. Let $i,j\in I$. We have then: $i\leqslant j$, by Fact 5.3, iff
$A^{+}_{i}\subseteq A^{+}_{j}$ iff $A_{f(i)}\subseteq A_{f(j)}$, by the
definition of partial order on the set of indexes, iff $f(i)\sqsubseteq f(j)$.

Ad 3. Conditions F1 and F2 are satisfied in an obvious way. By Fact 5.5 F3 is
also satisfied. ∎

Algebras from the frame of isolated algebras are not disjoint. In order to
obtain structures composed of algebras with disjoint universes, we will
consider the set difference of algebras from the frame and its proper
subalgebras.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$ be a frame
of algebras isolated wrt $\boldsymbol{A}$. For any $i\in I$, we put:

$A_{i}^{-}:=A_{i}\setminus\bigcup_{j<i}A_{j}.$

Note that by Corollary 10, for any $i\in I$, $A_{i}^{-}\neq\emptyset$ and
$A_{i}^{-}$ is closed under the operations of a given algebra
$\boldsymbol{A}$. For any $i\in I$, an algebra of the form:

$\boldsymbol{A}_{i}^{-}=\langle
A_{i}^{-};F_{1}^{\boldsymbol{A}_{i}^{-}},\allowbreak\ldots,F_{n}^{\boldsymbol{A}_{i}^{-}}\rangle$

we call a complement algebra (of $\boldsymbol{A}_{i}$).

The concept of the family of complement algebras is not sufficient to
decompose the Płonka sum. Two problems arise. The first is that we have not
specified how we should determine the values of operations on arguments from
different algebras of this family of complement algebras. The second one is
related to the problem of the existence of homomorphisms (see Appendix,
Example 31) — the definition of which is required by the definition of the
Płonka sum. We will solve both problems in the next section

###  3.2 Płonka Homomorphism

Płonka’s construction requires the existence of homomorphisms from algebras of
a given index to algebras of a ‘greater’ index, where the algebras have
disjoint universes. As shown in the example 31 (see Appendix), there is not
always a homomorphism from a complement algebra to a complement algebra with a
greater index. Moreover, not every homomorphism from a complement algebra to a
complement algebra is a homomorphism suitable for determining the operation of
a Płonka sum in the way specified in Definition 2. Such a situation is shown
in Example 32 (see Appendix).

Thus, we can conclude that having a given algebra that is a Płonka sum and
trying to identify the respective homomorphisms between complement algebras,
we distinguish three possibilities. The first is when homomorphisms do not
exist. Such a situation is presented in Example 31. The second is when
homomorphisms exist, but they do not satisfy the condition given in Definition
2. Such a situation is described in Example 32. Finally, the third, when
homomorphisms exist and they allow us to determine the operations of the
Płonka sum according to the mentioned condition. Such a situation has been
presented in Examples 31 and 32 (also in Examples 26 and 27).

Below, we introduce the general notion of a Płonka homomorphism, which will
allow us to identify all homomorphisms of the third possibility.

######  Definition 16.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra,
a family $\\{\boldsymbol{A}_{i}\\}_{i\in I}$ of subalgebras of
$\boldsymbol{A}$ indexed by a set $I$ partially oredered by $\leqslant$, and
let $i,j\in I$, where $i\leqslant j$. A function
$\varphi\in\mathsf{Hom}(\boldsymbol{A}_{i},\boldsymbol{A}_{j})$ is a _Płonka
homomorphism_ (_P-homomorphism_) _wrt $\boldsymbol{A}$_ iff one of the
following cases holds:

  * •

$i=j$ and $\varphi$ is an identity homomorphism,

  * •

$i<j$ and for any $m\leq n$ and $a_{1},\ldots,a_{ar(m)}\in A$:

$F^{\boldsymbol{A}}_{m}(a_{1},\ldots,\allowbreak
a_{ar(m)})=F^{\boldsymbol{A}}_{m}(f_{\varphi}(a_{1}),\ldots,\allowbreak
f_{\varphi}(a_{ar(m)})),$

if $F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{j\leqslant
k}A_{k}$.

The set of all P-homomorphisms wrt $\boldsymbol{A}$ between algebras
$\boldsymbol{A}_{i}$ and $\boldsymbol{A}_{j}$ is denoted by
$\mathsf{PHom}^{\boldsymbol{A}}(\boldsymbol{A}_{i},\boldsymbol{A}_{j})$.

Before we proceed with further analysis of P-homomorphism, let us make some
observations about the antecedent of the defining component of P-homomorphism
in the context of complement algebras.

######  Fact 17.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{m}^{\boldsymbol{A}}\rangle$ be an algebra
and let $\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$ be a
frame of algebras isolated wrt $\boldsymbol{A}$. Then, for all $m\leq n$, for
all $a_{1},\ldots,a_{ar(m)}\in A$, if there is $l\leq ar(m)$ s.t.
$a_{l}\in\bigcup_{i\leqslant j}A_{j}^{-}$, then
$F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{i\leqslant
j}A_{j}^{-}$.

###### Proof.

Let $m\leq n$ and $a_{1},\ldots,a_{ar(m)}\in A$. Assume that for $l_{0}\leq
ar(m)$ we have $a_{l_{0}}\in\bigcup_{i\leqslant j}A_{j}^{-}$. Let
$a_{l_{0}}\in A_{j_{0}}^{-}$, where $i\leqslant j_{0}$. Thus, by (3.1), if
$j<i$, then $a_{l_{0}}\notin A_{j}$. Similarly, if $j\nleqslant i\nleqslant
j$, then $a_{l_{0}}\notin A_{j}$. Assume that $j_{1}\nleqslant i\nleqslant
j_{1}$ and $a_{l_{0}}\in A_{j_{1}}$. Hence, $j_{0}\nleqslant j_{1}$,
$j_{1}\nleqslant j_{0}$ and $a_{l_{0}}\in A_{j_{0}}\cap A_{j_{1}}$. Since
$a_{l_{0}}\in A_{j_{0}}\cap A_{j_{1}}$, so by Fact 9, we have $j_{2}\in I$
s.t. $A_{j_{2}}=A_{j_{0}}\cap A_{j_{1}}$. However, $j_{2}<j_{0}$ and
$a_{l_{0}}\in A_{j_{2}}$. Therefore, $a_{l_{0}}\not\in A_{j_{0}}^{-}$.

Assume that
$F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\notin\bigcup_{i\leqslant
j}A_{j}^{-}$. Then, by (3.1), for any $j\in I$, if $i\leqslant j$, then
$F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\notin A_{j}^{-}$. However,
there is $j_{3}\in I$ s.t. $F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\in
A_{j_{3}}^{-}$, hence also $F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\in
A_{j_{3}}$. So, $j_{3}<i$ or $j_{3}\nleqslant i\nleqslant j_{3}$. Then,
$a_{l_{0}}\notin A_{j_{3}}$. Consequently, by Definition 3,
$F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{ar(m)})\notin A_{j_{3}}$. ∎

In general, the converse implication of the implication of Fact 17 doesn’t
have to hold.

Generalizing Example 33, we can say that any homomorphism of a direct system
is a P-homomorphism wrt the Płonka sum defined this direct system.

######  Fact 18.

Let $\mathcal{A}=\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in
I},\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$ be a direct system.
Then, for all $i,j\in I$, if $i\leqslant j$, then $\varphi_{i,j}$ is a
P-homomorphism wrt $S(\mathcal{A})$.

###### Proof.

Let $i_{0},j_{0}\in I$ and $i_{0}\leqslant j_{0}$. If $i_{0}=j_{0}$, then
$\varphi_{i_{0},j_{0}}$ is a P-homomorphism. Assume that $i_{0}<j_{0}$. Let
$m\leq n$ and $a_{1},\ldots,a_{ar(m)}\in\bigcup_{i\in I}A_{i}$. Assume that
$F^{S(\mathcal{A})}_{m}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{j_{0}\leqslant
k}A_{k}$.

Consider the case that there is no $l\leq ar(m)$ s.t. $a_{l}\in A_{i_{0}}$.
Then,
$F^{S(\mathcal{A})}_{m}(a_{1},\allowbreak\ldots,a_{ar(m)})\allowbreak=F^{S(\mathcal{A})}_{m}(f_{\varphi_{i_{0},j_{0}}}(a_{1}),\ldots,\allowbreak
f_{\varphi_{i_{0},j_{0}}}(a_{ar(m)}))$.

Consider the case that there is $l\leq ar(m)$ s.t. $a_{l}\in A_{i_{0}}$.
Without loss of generality we can assume that $l_{0}\neq ar(m)$,
$a_{1},\ldots,a_{l_{0}}\in A_{i_{0}}$ and $a_{l_{0}+1},\ldots,\allowbreak
a_{ar(m)}\notin A_{i_{0}}$.

By Definition 2,
$F^{S(\mathcal{A})}_{m}(a_{1},\ldots,a_{l_{0}},a_{l_{0}+1},\ldots\allowbreak
a_{ar(m)})=F^{\boldsymbol{A}_{k_{0}}}_{m}(\allowbreak\varphi_{i_{0},k_{0}}(\allowbreak
a_{1}),\allowbreak\ldots,\allowbreak\varphi_{i_{0},k_{0}}(\allowbreak
a_{l_{0}}),\varphi_{j_{l_{0}+1},k_{0}}(\allowbreak
a_{l_{0}+1}),\allowbreak\ldots,\varphi_{j_{ar(m)},k_{0}}(a_{ar(m)}))$, where
$i_{0}\vee j_{l_{0}+1}\vee\allowbreak\ldots\vee j_{ar(m)}=k_{0}$ and for any
$l_{0}+1\leq l\leq ar(m)$, $a_{l}\in A_{j_{l}}$.

By the initial assumption
$F^{S(\mathcal{A})}_{m}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{j_{0}\leqslant
k}A_{k}$, so $j_{0}\leqslant k_{0}$. Besides, since $k_{0}=i_{0}\vee
j_{l_{0}+1}\vee\ldots\vee j_{ar(m)}\leqslant j_{0}\vee
j_{l_{0}+1}\vee\ldots\vee j_{ar(m)}$, then $k_{0}=j_{0}\vee
j_{l_{0}+1}\vee\ldots\vee j_{ar(m)}$.

By Definition 1, for any $l\leq l_{0}$,
$\varphi_{i_{0},k_{0}}(a_{l})=\varphi_{j_{0},k_{0}}(\varphi_{i_{0},j_{0}}(a_{l}))$.
Thus, by Definition 1, we receive:

$\displaystyle F^{S(\mathcal{A})}_{m}(a_{1},\ldots,\allowbreak a_{ar(m)})=\,$
$\displaystyle
F^{\boldsymbol{A}_{k_{0}}}_{m}(\varphi_{i_{0},k_{0}}(a_{1}),\ldots,\varphi_{i_{0},k_{0}}(a_{l_{0}}),$
$\displaystyle\,\varphi_{j_{l_{0}+1},k_{0}}(a_{l_{0}+1}),\ldots,\varphi_{j_{ar(m)},k_{0}}(a_{ar(m)}))$
$\displaystyle=\,$ $\displaystyle
F^{\boldsymbol{A}_{k_{0}}}_{m}(\varphi_{j_{0},k_{0}}(\varphi_{i_{0},j_{0}}(a_{1})),\ldots,\varphi_{j_{0},k_{0}}(\varphi_{i_{0},j_{0}}(a_{l_{0}})),$
$\displaystyle\varphi_{j_{l_{0}+1},k_{0}}(a_{l_{0}+1}),\ldots,\varphi_{j_{ar(m)},k_{0}}(a_{ar(m)}))$
$\displaystyle=\,$ $\displaystyle
F^{S(\mathcal{A})}_{m}(\varphi_{i_{0},j_{0}}(a_{1}),\ldots,\varphi_{i_{0},j_{0}}(a_{l_{0}}),a_{l_{0}+1},\ldots,a_{ar(m)})$
$\displaystyle=\,$ $\displaystyle F^{S(\mathcal{A})}_{m}(\allowbreak
f_{\varphi_{i_{0},j_{0}}}(a_{1}),\ldots,f_{\varphi_{i_{0},j_{0}}}(a_{ar(m)})).$

Let us now turn to a discussion of the issue of the composition of
P-homomorphisms (cf. the conditions for the composition of homomorphisms given
in Definition 1). Let us first note that from a complement algebra to a
complement algebra with a greater index there can be more than one
P-homomorphism. Such a situation is shown in Example 34 (see Appendix). Below
we will show that the composition of any P-homomorphisms defined for a
complement algebra is a P-homomorphism. Let us start with the following fact.

######  Fact 19.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra,
$\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$ be a frame of
algebras isolated wrt $\boldsymbol{A}$,
$\varphi_{i,j}\in\mathsf{PHom}^{\boldsymbol{A}}(\boldsymbol{A}_{i}^{-},\boldsymbol{A}_{j}^{-})$
and
$\varphi_{j,k}\in\mathsf{PHom}^{\boldsymbol{A}}(\boldsymbol{A}_{j}^{-},\boldsymbol{A}_{k}^{-})$,
while $i,j,k\in I$ such that $i<j<k$. Then, for any $m\leq n$ and
$a_{1},\ldots,\allowbreak a_{ar(m)}\allowbreak\in\allowbreak A$, if
$F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{k\leqslant
p}A_{p}^{-}$, then
$F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}\allowbreak(f_{\varphi_{i,j}}\allowbreak(a_{1})),\ldots,\allowbreak
f_{\varphi_{j,k}}(\allowbreak
f_{\varphi_{i,j}}\allowbreak(a_{ar(m)})))=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{1}),\ldots,f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{ar(m)}))$.

###### Proof.

Let $a_{1},\ldots,a_{ar(m)}\in A$ and assume
$F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{k\leqslant
p}A_{p}^{-}$. Hence,
$F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{j\leqslant
p}A_{p}^{-}$.

We will show that there is $l\leq ar(m)$ s.t. $a_{l}\notin A_{i}^{-}$ and
$a_{l}\notin A_{j}^{-}$. Assume that for any $l\leq ar(m)$, $a_{l}\in
A^{-}_{i}$ or $a_{l}\in A^{-}_{j}$. Then,
$F^{\boldsymbol{A}}_{m}(f_{\varphi_{i,j}}(a_{1}),\ldots,\allowbreak
f_{\varphi_{i,j}}(a_{ar(m)}))\in A_{j}^{-}$. By Definition 16,
$F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})=F^{\boldsymbol{A}}_{m}(f_{\varphi_{i,j}}(a_{1}),\allowbreak\ldots,f_{\varphi_{i,j}}(a_{ar(m)}))$.
Therefore, by (3.1),
$F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\notin\bigcup_{k\leqslant
p}A_{p}^{-}$.

Consider the following cases:

  * a.

there is no $l\leq ar(m)$ s.t. $a_{l}\in A_{i}^{-}$ and there is no $l\leq
ar(m)$ s.t. $a_{l}\in A_{j}^{-}$,

  * b.

there is no $l\leq ar(m)$ s.t. $a_{l}\in A_{i}^{-}$ and there is $l\leq ar(m)$
s.t. $a_{l}\in A_{j}^{-}$,

  * c.

there is $l\leq ar(m)$ s.t. $a_{l}\in A_{i}^{-}$ and there is no $l\leq ar(m)$
s.t. $a_{l}\in A_{j}^{-}$,

  * d.

there is $l\leq ar(m)$ s.t. $a_{l}\in A_{i}^{-}$ and there is $l\leq ar(m)$
s.t. $a_{l}\in A_{j}^{-}$.

Case a. By the definition of the function $f_{\varphi}$ we get:

$\displaystyle
F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{1})),\ldots,f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{ar(m)})))=$
$\displaystyle=F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}\circ\,f_{\varphi_{i,j}}(a_{1}),\ldots,f_{\varphi_{j,k}}\circ\,f_{\varphi_{i,j}}(a_{ar(m)})).$

Case b. Without loss of generality we assume that only
$a_{1},\allowbreak\ldots,a_{l}\in A_{j}^{-}$, where $l\lneq ar(m)$. By the
definition of the function $f_{\varphi}$ and by Definition 16 we receive:

$\displaystyle
F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{1})),\ldots,f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{ar(m)})))=$
$\displaystyle=F^{\boldsymbol{A}}_{m}(\varphi_{j,k}(a_{1}),\ldots,\varphi_{j,k}(a_{l}),a_{l+1},\ldots,a_{ar(m)})$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}(a_{1}),\ldots,f_{\varphi_{j,k}}(a_{l}),f_{\varphi_{j,k}}(a_{l+1}),\ldots,f_{\varphi_{j,k}}(a_{ar(m)}))$
$\displaystyle=F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{l},a_{l+1},\ldots,a_{ar(m)})$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{1}),\ldots,f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{l}),f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{l+1}),\ldots,f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{ar(m)})).$

Case c. Without loss of generality, we assume that only
$a_{1},\allowbreak\ldots,a_{l}\in A_{i}$, where $l\lneq ar(m)$. By the
definition of the function $f_{\varphi}$ we get:

$\displaystyle
F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{1})),\ldots,f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{ar(m)})))=$
$\displaystyle=F^{\boldsymbol{A}}_{m}(\varphi_{j,k}(\varphi_{i,j}(a_{1})),\ldots,\varphi_{j,k}(\varphi_{i,j}(a_{l})),a_{l+1},\ldots,a_{ar(m)})$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{1}),\ldots,f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{l}),f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{l+1}),\ldots,f_{\varphi_{j,k\circ\,\varphi_{i,j}}}(a_{ar(m)})).$

Case d. Without loss of generality, we assume that only
$a_{1},\allowbreak\ldots,a_{l}\in A_{j}^{-}$ and $a_{l+1},\ldots,a_{l+p}\in
A_{i}^{-}$, where $l+p\lneq ar(m)$. Notice that
$\varphi_{j,k}(\varphi_{i,j}(a_{l+1})),\ldots,\allowbreak\varphi_{j,k}(\varphi_{i,j}(a_{l+p}))\in\bigcup_{k\leqslant
p}A^{-}_{p}$. By the definition of the function $f_{\varphi}$, Fact 17 and
Definition 16, we receive:

$\displaystyle\,F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{1})),\ldots,f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{ar(m)})))=$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}(a_{1}),\ldots,f_{\varphi_{j,k}}(a_{l}),\varphi_{j,k}(\varphi_{i,j}(a_{l+1})),\ldots,\varphi_{j,k}(\varphi_{i,j}(a_{l+p})),$
$\displaystyle\;\;\;\;\;f_{\varphi_{j,k}}(a_{l+p+1}),\ldots,f_{\varphi_{j,k}}(a_{ar(m)}))$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}(a_{1}),\ldots,f_{\varphi_{j,k}}(a_{l}),f_{\varphi_{j,k}}(\varphi_{j,k}(\varphi_{i,j}(a_{l+1}))),\ldots,f_{\varphi_{j,k}}(\varphi_{j,k}(\varphi_{i,j}(a_{l+p}))),$
$\displaystyle\;\;\;\;\;f_{\varphi_{j,k}}(a_{l+p+1}),\ldots,f_{\varphi_{j,k}}(a_{ar(m)}))$
$\displaystyle=F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{l},\varphi_{j,k}(\varphi_{i,j}(a_{l+1})),\ldots,\varphi_{j,k}(\varphi_{i,j}(a_{l+p})),$
$\displaystyle\;\;\;\;\;a_{l+p+1},\ldots,a_{ar(m)})$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{1}),\ldots,f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{l}),f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{l+1}),\ldots,f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{l+p}),$
$\displaystyle\;\;\;\;\;f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{l+p+1}),\ldots,f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{ar(m)})).\hskip
167.87108pt\qed$

Now we can easily prove the following fact concerning the composition of
P-homomorphisms.

######  Fact 20.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$ be a frame
of algebras isolated wrt $\boldsymbol{A}$. Then, for all $i,j,k\in I$, if
$i\leqslant j\leqslant k$,
$\varphi_{i,j}\in\mathsf{PHom}^{\boldsymbol{A}}(\boldsymbol{A}_{i}^{-},\boldsymbol{A}_{j}^{-})$
and
$\varphi_{j,k}\in\mathsf{PHom}^{\boldsymbol{A}}(\boldsymbol{A}_{j}^{-},\boldsymbol{A}_{k}^{-})$,
then
$\varphi_{j,k}\circ\varphi_{i,j}\in\mathsf{PHom}^{\boldsymbol{A}}(\boldsymbol{A}_{i}^{-},\boldsymbol{A}_{k}^{-})$.

###### Proof.

Assume that $i<j<k$. In the other cases, the composition equals one of the
given P-homomorphisms.

Let $a_{1},\ldots,a_{ar(m)}\in A$. Assume that
$F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{k\leqslant
p}A_{p}^{-}$. Hence,
$F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{j\leqslant
p}A_{p}^{-}$. Be Definition 16,
$F^{\boldsymbol{A}}_{m}(f_{\varphi_{i,j}}(a_{1}),\allowbreak\ldots,\allowbreak
f_{\varphi_{i,j}}(a_{ar(m)}))=F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\in\bigcup_{k\leqslant
p}A_{p}^{-}$. Therefore, by Definition 16 and Fact 19 we obtain:

$\displaystyle F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{m})$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{i,j}}(a_{1}),\ldots,f_{\varphi_{i,j}}(a_{m}))$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{1})),\ldots,f_{\varphi_{j,k}}(f_{\varphi_{i,j}}(a_{m})))$
$\displaystyle=F^{\boldsymbol{A}}_{m}(f_{\varphi_{j,k}\circ\,\varphi_{i,j}}(a_{1}),\ldots,f_{\varphi_{j,k}\circ\,f_{\varphi_{i,j}}}(a_{m})).$

In the set of all frames of isolated algebras, we have to distinguish those
frames, for which from any complement algebra to any complement algebra of a
greater index, there is a P-homomorphism. Moreover, such homomorphisms have to
satisfy the conditions for the composition of homomorphisms given in
Definition 1. To this aim, we introduce the notion of a sound set of
P-homomorphisms.

######  Definition 21.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra
and $\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$ be a frame
of algebras isolated wrt $\boldsymbol{A}$. A sound set of P-homomorphisms is a
set $\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}$ s.t. for any $i,j\in I$:

  * •

if $i=j$ or $i\prec j$, then
$\varphi_{i,j}\in\mathsf{PHom}^{\boldsymbol{A}}(\boldsymbol{A}_{i}^{-},\boldsymbol{A}_{j}^{-})$,

  * •

if $i<j$ and $i\nprec j$, then the following conditions are satisfied:

    * –

$\varphi_{i,j}=\varphi_{i_{m},j}\circ\ldots\circ\varphi_{i,i_{1}}$, for some
$i_{1},\ldots,i_{m}\in I$ s.t. $i\prec i_{1}\prec\ldots\prec i_{m}\prec j$,

    * –

for any $j_{1},\ldots,j_{o}\in I$, if $i\prec j_{1}\prec\ldots\prec j_{o}\prec
j$, then $\varphi_{i,j}=\varphi_{j_{o},j}\circ\ldots\circ\varphi_{i,j_{1}}$.

##  4 Decomposition Theorem

Let us now move on to the last construction.

######  Definition 22.

Let $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ be an algebra.

  * •

A _system of algebras isolated wrt $\boldsymbol{A}$_ is a triple
$\langle\mathcal{I},\\{\boldsymbol{A}_{i}^{-}\\}_{i\in
I},\allowbreak\\{\varphi_{i,j}\\}_{i\leqslant j;\allowbreak i,j\in I}\rangle$
s.t. $\langle\mathcal{I},\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$ is a frame
of algebras isolated wrt $\boldsymbol{A}$ and
$\\{\varphi_{i,j}\allowbreak\\}_{i\leqslant j;i,j\in I}$ is a sound set of
P-homomorphisms.

  * •

We call a system of isolated algebras
$\langle\mathcal{I},\\{\boldsymbol{A}_{i}^{-}\\}_{i\in
I},\allowbreak\\{\varphi_{i,j}\\}_{i\leqslant j;\allowbreak i,j\in I}\rangle$
a non-trivial one iff there are $i,j\in I$ s.t. $i\neq j$.

Let us now show that the set of systems of isolated algebras is non-empty.

######  Theorem 23.

If $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ is Płonka sum,
then there exists system of algebras isolated wrt $\boldsymbol{A}$.

###### Proof.

Let $\mathcal{A}=\langle\langle
I,\leqslant\rangle,\\{\boldsymbol{A}_{i}\\}_{i\in
I},\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$ be a direct system s.t.
$\boldsymbol{A}=S(\mathcal{A})$. Let $\langle\langle
J,\sqsubseteq\rangle,\\{\boldsymbol{B}_{j}\\}_{j\in J}\rangle$ is a frame of
algebras isolated wrt $\boldsymbol{A}$, where $\\{\boldsymbol{B}_{j}\\}_{j\in
J}$ is the set of all isolated algebras wrt $\boldsymbol{A}$.

By Fact 15, we have injection $f\colon I\longrightarrow J$ defined by the
formula $f(i):=j$, where $A^{+}_{i}=B_{j}$. We have also that $\langle\langle
I,\leqslant\rangle,\\{\boldsymbol{A}_{i}^{+}\\}_{i\in I}\rangle$ and
$\langle\langle
f(I),\allowbreak\sqsubseteq\rangle,\allowbreak\\{\boldsymbol{B}_{k}\\}_{k\in
f(I)}\rangle$ are isomorphic. Therefore, isomorphic are also $\langle\langle
I,\leqslant\rangle,\\{\boldsymbol{A}_{i}\\}_{i\in I}\rangle$ and
$\langle\langle f(I),\sqsubseteq\rangle,\\{\boldsymbol{B}_{k}^{-}\\}_{k\in
f(I)}\rangle$.

For all $j_{1},j_{2}\in J$, for every $a\in B_{j_{1}}$ we put
$\psi_{j_{1},j_{2}}(a):=\varphi_{i_{1},i_{2}}(a)$, where $j_{1}=f(i_{1})$ and
$j_{2}=f(i_{2})$. By Fact 18 and Definition 1, $\\{\psi_{j,k}\\}_{j\sqsubseteq
k;j,k\in f(I)}$ is a sound set of P-homomorphisms.

So, $\langle\langle f(I),\sqsubseteq\rangle,\\{\boldsymbol{B}_{k}\\}_{k\in
f(I)},\\{\psi_{j,k}\\}_{j\sqsubseteq k;j,k\in f(I)}\rangle$ is a system of
algebras isolated wrt $\boldsymbol{A}$. ∎

The following theorem shows that isolated algebras allow us to decompose the
Płonka sum.

######  Theorem 24 (Decomposition theorem).

If $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ is an algebra
and
$\mathcal{A}=\langle\mathcal{I},\allowbreak\\{\boldsymbol{A}_{i}^{-}\\}_{i\in
I},\allowbreak\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$ is a non-
trivial system of algebras isolated wrt $\boldsymbol{A}$, then $\mathcal{A}$
is a direct system and $\boldsymbol{A}=S(\mathcal{A})$.

###### Proof.

Obviously, $\mathcal{A}$ is a non-trivial direct system.

Let $m\leq n$ and $a_{1},\ldots,a_{ar(m)}\in A$. Assume that for any $l\leq
ar(m)$, $a_{l}\in A^{-}_{i_{l}}$ and $j=i_{1}\vee\ldots\vee i_{ar(m)}$. Thus,
for any $l\leq ar(m)$, $\varphi_{i_{l},j}a_{l}\in A^{-}_{j}$. Since for any
$l\leq ar(m)$, $a_{l}\in A^{-}_{i_{l}}$, then for any $l\leq ar(m)$, $a_{l}\in
A_{j}$. Hence, $F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\in A_{j}$.
Moreover,
$F_{m}^{\boldsymbol{A}}(\varphi_{i_{1,j}}(a_{1}),\ldots,a_{l},\ldots,a_{ar(m)}),\allowbreak\ldots,F_{m}^{\boldsymbol{A}}(\varphi_{i_{1},j}(a_{1}),\ldots,\varphi_{i_{l},j}(a_{l}),\allowbreak\ldots,\allowbreak
a_{ar(m)}),\ldots,F_{m}^{\boldsymbol{A}}(\varphi_{i_{1},j}(a_{1}),\allowbreak\ldots,\allowbreak\varphi_{i_{l},j}(a_{l}),\allowbreak$
$\ldots,\allowbreak\varphi_{i_{ar(m)},j}(a_{ar(m)}))\in A_{j}$.

Assume indirectly that there is $k<j$ s.t.
$F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\in A_{k}$. By Definition 3,
$a_{1},\ldots,a_{ar(m)}\in A_{k}$. By Fact 5.5, for any $l\leq ar(m)$,
$i_{l}\leqslant k$. So, $j\leqslant k$. Similarly we can show that there is no
$k<j$ s.t. at least one of the following holds:
$F_{m}^{\boldsymbol{A}}(\varphi_{i_{1},j}(a_{1}),\ldots,a_{l},\ldots,a_{ar(m)})\in
A_{k},\ldots,F_{m}^{\boldsymbol{A}}(\varphi_{i_{1},j}(a_{1}),\ldots,\allowbreak\varphi_{i_{l},j}(a_{l}),\allowbreak\ldots,\allowbreak
a_{ar(m)})\in
A_{k},\ldots,F_{m}^{\boldsymbol{A}}(\varphi_{i_{1},j}(a_{1}),\ldots,\varphi_{i_{l},j}(a_{l}),\ldots,\allowbreak\varphi_{i_{ar(m)},j}(\allowbreak
a_{ar(m)}\allowbreak))\allowbreak\in A_{k}$.

Thus, by (3.1), $F^{\boldsymbol{A}}_{m}(a_{1},\ldots,a_{ar(m)})\in
A_{j}^{-}\subseteq\bigcup_{j\leqslant k}A_{k}^{-}$. And also:
$F_{m}^{\boldsymbol{A}}(\varphi_{i_{1},j}(\allowbreak
a_{1}),\allowbreak\ldots,\allowbreak a_{l},\ldots,\allowbreak a_{ar(m)})\in
A_{j}^{-},\allowbreak\ldots,\allowbreak
F_{m}^{\boldsymbol{A}}(\varphi_{i_{1}},j(a_{1}),\allowbreak\ldots,\allowbreak\varphi_{i_{l},j}(a_{l}),\allowbreak\ldots,\allowbreak
a_{ar(m)})\allowbreak\\!\in\\!\allowbreak
A_{j}^{-},\allowbreak\ldots,\allowbreak
F_{m}^{\boldsymbol{A}}(\varphi_{i_{1}},j(a_{1}),\ldots,\varphi_{i_{l},j}(a_{l}),\ldots,\allowbreak\varphi_{i_{ar(m)},j}(\allowbreak
a_{ar(m)}\allowbreak))\allowbreak\in A_{j}^{-}$.

By the definition of $\leqslant$ and by (3.1), for all $o,p\leq ar(m)$, if
$o\neq p$, then $A_{i_{o}}^{-}\neq A_{i_{p}}^{-}$

Hence, by Definition 16, the definition of $f_{\varphi}$ and Fact 17, we
obtain:

$\displaystyle F_{m}^{\boldsymbol{A}}(a_{1},\ldots,a_{l},\ldots,a_{ar(m)})=$
$\displaystyle
F_{m}^{\boldsymbol{A}}(f_{\varphi_{i_{1},j}}(a_{1}),\ldots,f_{\varphi_{i_{1},j}}(a_{l}),\ldots,f_{\varphi_{i_{1},j}}(a_{ar(m)}))$
$\displaystyle=\;$ $\displaystyle
F_{m}^{\boldsymbol{A}}(\varphi_{i_{1},j}(a_{1}),\ldots,a_{l},\ldots,a_{ar(m)})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\vdots$ $\displaystyle=\;$ $\displaystyle
F_{m}^{\boldsymbol{A}}(f_{\varphi_{i_{l},j}}(\varphi_{i_{1},j}(a_{1})),\ldots,f_{\varphi_{i_{l},j}}(a_{l}),\ldots,f_{\varphi_{i_{l},j}}(a_{ar(m)}))$
$\displaystyle=\;$ $\displaystyle
F_{m}^{\boldsymbol{A}}(\varphi_{i_{1},j}(a_{1}),\ldots,\varphi_{i_{l},j}(a_{l}),\ldots,a_{ar(m)})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\vdots$ $\displaystyle=$
$\displaystyle\;F_{m}^{\boldsymbol{A}}(\varphi_{i_{1},j}(a_{1}),\ldots,\varphi_{i_{l},j}(a_{l}),\ldots,\varphi_{i_{ar(m)},j}(a_{ar(m)})$
$\displaystyle=$
$\displaystyle\;F_{m}^{\boldsymbol{A}_{j}}(\varphi_{i_{1},j}(a_{1}),\ldots,\varphi_{i_{l},j}(a_{l}),\ldots,\varphi_{i_{ar(m)},j}(a_{ar(m)}).\hskip
17.07182pt\qed$

By Theorems 23 and 24 the following conclusion follows:

######  Corollary 25.

Algebra $\boldsymbol{A}=\langle
A;F_{1}^{\boldsymbol{A}},\ldots,F_{n}^{\boldsymbol{A}}\rangle$ is a Płonka sum
iff there is a non-trivial system of algebras isolated wrt $\boldsymbol{A}$.

##  5 Two Examples of Decompositions

In this section, we will apply our decomposition method to two algebras. We
will show how, we are able to determine all (up to isomorphism) direct systems
of a given Płonka sum. As well as to prove that a given algebra is not a
Płonka sum if it is impossible to determine a system of algebras isolated wrt
this algebra.

Let us consider two algebras:

  * •

$\boldsymbol{A}_{7}=\langle\\{a_{1},\ldots,a_{11}\\};\ast\rangle$,

  * •

$\boldsymbol{A}_{8}=\langle\\{a_{1},\ldots,a_{10}\\};\ast\rangle$,

with binary operations defined by Tables 1 and 2, respectively.

$\ast$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | $a_{6}$ | $a_{7}$ | $a_{8}$ | $a_{9}$ | $a_{10}$ | $a_{11}$
---|---|---|---|---|---|---|---|---|---|---|---
$a_{1}$ | $a_{2}$ | $a_{1}$ | $a_{6}$ | $a_{5}$ | $a_{6}$ | $a_{5}$ | $a_{8}$ | $a_{7}$ | $a_{9}$ | $a_{11}$ | $a_{10}$
$a_{2}$ | $a_{2}$ | $a_{1}$ | $a_{6}$ | $a_{5}$ | $a_{6}$ | $a_{5}$ | $a_{8}$ | $a_{7}$ | $a_{9}$ | $a_{11}$ | $a_{10}$
$a_{3}$ | $a_{6}$ | $a_{5}$ | $a_{4}$ | $a_{3}$ | $a_{6}$ | $a_{5}$ | $a_{8}$ | $a_{7}$ | $a_{9}$ | $a_{11}$ | $a_{10}$
$a_{4}$ | $a_{6}$ | $a_{5}$ | $a_{4}$ | $a_{3}$ | $a_{6}$ | $a_{5}$ | $a_{8}$ | $a_{7}$ | $a_{9}$ | $a_{11}$ | $a_{10}$
$a_{5}$ | $a_{6}$ | $a_{5}$ | $a_{6}$ | $a_{5}$ | $a_{6}$ | $a_{5}$ | $a_{8}$ | $a_{7}$ | $a_{9}$ | $a_{11}$ | $a_{10}$
$a_{6}$ | $a_{6}$ | $a_{5}$ | $a_{6}$ | $a_{5}$ | $a_{6}$ | $a_{5}$ | $a_{8}$ | $a_{7}$ | $a_{9}$ | $a_{11}$ | $a_{10}$
$a_{7}$ | $a_{8}$ | $a_{7}$ | $a_{8}$ | $a_{7}$ | $a_{8}$ | $a_{7}$ | $a_{8}$ | $a_{7}$ | $a_{10}$ | $a_{11}$ | $a_{10}$
$a_{8}$ | $a_{8}$ | $a_{7}$ | $a_{8}$ | $a_{7}$ | $a_{8}$ | $a_{7}$ | $a_{8}$ | $a_{7}$ | $a_{10}$ | $a_{11}$ | $a_{10}$
$a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{10}$ | $a_{11}$ | $a_{9}$ | $a_{11}$ | $a_{10}$
$a_{10}$ | $a_{10}$ | $a_{11}$ | $a_{10}$ | $a_{11}$ | $a_{10}$ | $a_{11}$ | $a_{10}$ | $a_{11}$ | $a_{10}$ | $a_{11}$ | $a_{10}$
$a_{11}$ | $a_{10}$ | $a_{11}$ | $a_{10}$ | $a_{11}$ | $a_{10}$ | $a_{11}$ | $a_{10}$ | $a_{11}$ | $a_{10}$ | $a_{11}$ | $a_{10}$
Table 1: Operation $\ast$ of algebra $\boldsymbol{A}_{7}$ $\ast$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | $a_{6}$ | $a_{7}$ | $a_{8}$ | $a_{9}$ | $a_{10}$
---|---|---|---|---|---|---|---|---|---|---
$a_{1}$ | $a_{1}$ | $a_{3}$ | $a_{3}$ | $a_{3}$ | $a_{6}$ | $a_{6}$ | $a_{8}$ | $a_{8}$ | $a_{9}$ | $a_{9}$
$a_{2}$ | $a_{3}$ | $a_{2}$ | $a_{3}$ | $a_{3}$ | $a_{6}$ | $a_{6}$ | $a_{8}$ | $a_{8}$ | $a_{9}$ | $a_{9}$
$a_{3}$ | $a_{3}$ | $a_{3}$ | $a_{4}$ | $a_{3}$ | $a_{5}$ | $a_{5}$ | $a_{7}$ | $a_{7}$ | $a_{10}$ | $a_{10}$
$a_{4}$ | $a_{3}$ | $a_{3}$ | $a_{3}$ | $a_{3}$ | $a_{5}$ | $a_{5}$ | $a_{7}$ | $a_{7}$ | $a_{10}$ | $a_{10}$
$a_{5}$ | $a_{6}$ | $a_{6}$ | $a_{5}$ | $a_{5}$ | $a_{5}$ | $a_{5}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$
$a_{6}$ | $a_{6}$ | $a_{6}$ | $a_{5}$ | $a_{5}$ | $a_{5}$ | $a_{5}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$
$a_{7}$ | $a_{8}$ | $a_{8}$ | $a_{7}$ | $a_{7}$ | $a_{9}$ | $a_{9}$ | $a_{7}$ | $a_{7}$ | $a_{9}$ | $a_{9}$
$a_{8}$ | $a_{8}$ | $a_{8}$ | $a_{7}$ | $a_{7}$ | $a_{9}$ | $a_{9}$ | $a_{7}$ | $a_{7}$ | $a_{9}$ | $a_{9}$
$a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{10}$ | $a_{9}$
$a_{10}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$ | $a_{9}$
Table 2: Operation $\ast$ of algebra $\boldsymbol{A}_{8}$

Our decomposition can be performed in two steps. The first step consists of
three components. First, we determine the family of algebras isolated wrt a
given algebra. Secondly, we select all frames of isolated algebras. Thirdly,
we modify the obtained frames by replacing the algebras of the given frame
with complement algebras. In the second step, we try to verify if a family of
P-homomorphisms can be selected.

Step 1.1. Let us determine all algebras isolated wrt algebras we are
considering.

  1. 1.

In the case of algebra $\boldsymbol{A}_{7}$ we have the following isolated
algebras:

     * •

$\boldsymbol{A}_{7}^{1}$ with the universe $\\{a_{1},a_{2}\\}$,

     * •

$\boldsymbol{A}_{7}^{2}$ with the universe $\\{a_{3},a_{4}\\}$,

     * •

$\boldsymbol{A}_{7}^{3}$ with the universe
$\\{a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\\}$,

     * •

$\boldsymbol{A}_{7}^{4}$ with the universe
$\\{a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7},a_{8}\\}$,

     * •

$\boldsymbol{A}_{7}^{5}$ with the universe
$\\{a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{9}\\}$,

     * •

$\boldsymbol{A}_{7}^{6}$ with the universe $\\{a_{1},\ldots,a_{11}\\}$.

  2. 2.

In the case of algebra $\boldsymbol{A}_{8}$ we have the following isolated
algebras:

     * •

$\boldsymbol{A}_{8}^{1}$ with the universe $\\{a_{1}\\}$,

     * •

$\boldsymbol{A}_{8}^{2}$ with the universe $\\{a_{2}\\}$,

     * •

$\boldsymbol{A}_{8}^{3}$ with the universe $\\{a_{1},a_{2},a_{3},a_{4}\\}$,

     * •

$\boldsymbol{A}_{8}^{4}$ with the universe
$\\{a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\\}$,

     * •

$\boldsymbol{A}_{8}^{5}$ with the universe
$\\{a_{1},a_{2},a_{3},a_{4},a_{7},a_{8}\\}$,

     * •

$\boldsymbol{A}_{8}^{6}$ with the universe $\\{a_{1},\ldots,a_{10}\\}$.

Let us adopt the following notation:
${\boldsymbol{A}_{n}^{m}}^{-X}:=\langle{A_{n}^{m}}^{-X},\ast\rangle$, where
$X\subseteq\\{i\in\mathbb{N}:i<m\\}$ and
${A^{m}_{n}}^{-X}=A^{m}_{n}\setminus\bigcup_{i\in X}A^{i}_{n}$.

Step 1.2. In Figure 1, we present all frames of algebras isolated wrt
$\boldsymbol{A}_{7}$ and $\boldsymbol{A}_{8}$, respectively.

$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(1)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(2)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{6}_{n}$(3)$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(4)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(5)$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(6)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(7)$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{6}_{n}$(8)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{6}_{n}$(9)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{6}_{n}$(10)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(11)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{6}_{n}$(12)$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{6}_{n}$(13)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{6}_{n}$(14)$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(15)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(16)$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(17)$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{6}_{n}$(18)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{6}_{n}$(19)$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{6}_{n}$(20)$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(21)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{6}_{n}$(22)$\boldsymbol{A}^{2}_{n}$$\boldsymbol{A}^{6}_{n}$(23)$\boldsymbol{A}^{1}_{n}$$\boldsymbol{A}^{6}_{n}$(24)$\boldsymbol{A}^{3}_{n}$$\boldsymbol{A}^{6}_{n}$(25)$\boldsymbol{A}^{5}_{n}$$\boldsymbol{A}^{6}_{n}$(26)$\boldsymbol{A}^{4}_{n}$$\boldsymbol{A}^{6}_{n}$(27)
Figure 1: Frames of algebras isolated wrt $\boldsymbol{A}_{7}$ and
$\boldsymbol{A}_{8}$

Step 1.3. Considering any frame presented in Figure 1, we replace any algebra
$\boldsymbol{A}_{n}^{m}$ in this frame with algebra
${\boldsymbol{A}_{n}^{m}}^{-}$, where
${A_{n}^{m}}^{-}={A_{n}^{m}}\setminus\bigcup_{i<m}A_{n}^{i}$.

Step 2. We proceed to the analysis of the possibility of determining
P-homomorphisms.

Let us start with algebra $\boldsymbol{A}_{7}$. We have the following
P-homomorphisms:

  * •

$\varphi_{1,3}(a_{1})=a_{5}$ and $\varphi_{1,3}(a_{2})=a_{6}$,
$\varphi_{1,4}(a_{1})=a_{7}$ and $\varphi_{1,4}(a_{2})=a_{8}$,
$\varphi_{1,6}(a_{1})=a_{11}$ and $\varphi_{1,6}(a_{2})=a_{10}$;

  * •

$\varphi_{2,3}(a_{3})=a_{5}$ and $\varphi_{2,3}(a_{4})=a_{6}$,
$\varphi_{2,4}(a_{3})=a_{7}$ and $\varphi_{2,4}(a_{4})=a_{8}$,
$\varphi_{2,6}(a_{3})=a_{11}$ and $\varphi_{2,6}(a_{4})=a_{10}$;

  * •

$\varphi_{3,4}(a_{5})=a_{7}$ and $\varphi_{3,4}(a_{6})=a_{8}$,
$\varphi_{3,6}(a_{5})=a_{11}$ and $\varphi_{3,6}(a_{6})=a_{10}$;

  * •

$\varphi_{4,6}(a_{7})=a_{11}$ and $\varphi_{4,6}(a_{8})=a_{12}$,

where
$\varphi_{i,j}\in\mathsf{Hom}({\boldsymbol{A}_{7}^{i}}^{-},{\boldsymbol{A}_{7}^{j}}^{-})$,
for the considered indexes $i,j$.

Note that there is no homomorphism from ${\boldsymbol{A}_{7}^{5}}^{-}$ to
${\boldsymbol{A}_{7}^{6}}\setminus\boldsymbol{A}_{7}^{5}$. By Table 1,
$a_{9}\ast a_{9}=a_{9}$. However, for every $a\in A_{7}^{6}\setminus
A_{7}^{5}$, $a\ast a\neq a$. Thus, frames (1), (2), (4), (5), (6), (7), (11),
(14), (15), (16), (21) and (26) form Figure 1, are not those over which we can
determine systems of algebras isolated wrt $\boldsymbol{A}_{7}$. In the case
of the remaining frames from Figure 1, the sound sets of P-homomorphisms can
be determined.

Possible to define systems of algebras isolated wrt $\boldsymbol{A}_{7}$ we
present in Figure 2. By Corollary 25 algebra $\boldsymbol{A}_{7}$ is a Płonka
sum.

Let us analyze algebra $\boldsymbol{A}_{8}$. We will consider case (1) from
Figure 1 and check whether there are P-homomorphisms from
$\boldsymbol{A}_{8}^{1}$ and $\boldsymbol{A}_{8}^{2}$ to algebras $\
A_{8}^{n^{-}}$, for any $2<n\leq 6$.

  * •

There are no homomorphisms from $\boldsymbol{A}_{8}^{1^{-}}$ and from
$\boldsymbol{A}_{8}^{2^{-}}$ to $\boldsymbol{A}_{8}^{3^{-}}$. Similarly, there
are no homomorphisms $\boldsymbol{A}_{8}^{1^{-}}$ and from
$\boldsymbol{A}_{8}^{2^{-}}$ to $\boldsymbol{A}_{8}^{6^{-}}$. By Table 2,
$a_{1}\ast a_{1}=a_{1}$ and $a_{2}\ast a_{2}=a_{2}$. In turn, $a_{3}\ast
a_{3}=a_{4}$ and $a_{4}\ast a_{4}=a_{3}$. An analogous dependence occurs when
trying to determine the homomorphism from $\boldsymbol{A}_{8}^{1^{-}}$ and
from $\boldsymbol{A}_{8}^{2^{-}}$ to $\boldsymbol{A}_{8}^{6^{-}}$.

  * •

There is one homomorphism from $\boldsymbol{A}_{8}^{1^{-}}$ and one from
$\boldsymbol{A}_{8}^{2^{-}}$ to $\boldsymbol{A}_{8}^{4^{-}}$. Similarly, there
is one homomorphism from $\boldsymbol{A}_{8}^{1^{-}}$ and from
$\boldsymbol{A}_{8}^{2^{-}}$ to $\boldsymbol{A}_{8}^{5^{-}}$. By Table 2,
$a_{5}\ast a_{5}=a_{5}$. However, such homomorphisms are not P-homomorphisms.
By Table 2, $a_{1}\ast a_{5}=a_{6}\neq a_{5}\ast a_{5}$ and $a_{2}\ast
a_{5}=a_{6}\neq a_{5}\ast a_{5}$. An analogous dependence occurs when
considering homomorphisms from $\boldsymbol{A}_{8}^{1^{-}}$ and from
$\boldsymbol{A}_{8}^{2^{-}}$ to $\boldsymbol{A}_{8}^{6^{-}}$.

According to our analysis, by Corollary 25, algebra $\boldsymbol{A}_{8}$ is
not a Płonka sum.

$\boldsymbol{A}^{1}_{7}$$\boldsymbol{A}^{2}_{7}$$\varphi_{1,3}$$\varphi_{2,3}$${\boldsymbol{A}^{3}_{7}}^{-\\{1,2\\}}$$\varphi_{3,4}$${\boldsymbol{A}_{7}^{4}}^{-\\{3\\}}$$\varphi_{4,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{4\\}}$(1)$\boldsymbol{A}_{7}^{2}$$\varphi_{2,3}$${\boldsymbol{A}_{7}^{3}}^{-\\{2\\}}$$\varphi_{3,4}$${\boldsymbol{A}_{7}^{4}}^{-\\{3\\}}$$\varphi_{4,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{4\\}}$(2)$\boldsymbol{A}_{7}^{1}$$\varphi_{1,3}$${\boldsymbol{A}_{7}^{3}}^{-\\{1\\}}$$\varphi_{3,4}$${\boldsymbol{A}_{7}^{4}}^{-\\{3\\}}$$\varphi_{4,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{4\\}}$(3)$\boldsymbol{A}_{7}^{1}$$\boldsymbol{A}_{7}^{2}$$\varphi_{1,3}$$\varphi_{2,3}$${\boldsymbol{A}_{7}^{3}}^{-\\{1,2\\}}$$\varphi_{3,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{3\\}}$(4)$\boldsymbol{A}_{7}^{1}$$\boldsymbol{A}_{7}^{2}$$\varphi_{1,4}$$\varphi_{2,4}$${\boldsymbol{A}_{7}^{4}}^{-\\{1,2\\}}$$\varphi_{4,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{4\\}}$(5)$\boldsymbol{A}_{7}^{2}$$\varphi_{2,3}$${\boldsymbol{A}_{7}^{3}}^{-\\{2\\}}$$\varphi_{3,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{3\\}}$(6)$\boldsymbol{A}_{7}^{1}$$\varphi_{1,3}$${\boldsymbol{A}_{7}^{3}}^{-\\{1\\}}$$\varphi_{3,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{3\\}}$(7)$\boldsymbol{A}_{7}^{2}$$\varphi_{2,4}$${\boldsymbol{A}_{7}^{4}}^{-\\{2\\}}$$\varphi_{4,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{4\\}}$(8)$\boldsymbol{A}_{7}^{1}$$\varphi_{1,4}$${\boldsymbol{A}_{7}^{4}}^{-\\{1\\}}$$\varphi_{4,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{4\\}}$(9)$\boldsymbol{A}_{7}^{3}$$\varphi_{3,4}$${\boldsymbol{A}_{7}^{4}}^{-\\{3\\}}$$\varphi_{4,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{4\\}}$(10)$\boldsymbol{A}_{7}^{1}$$\boldsymbol{A}_{7}^{2}$$\varphi_{1,6}$$\varphi_{2,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{1,2\\}}$(11)$\boldsymbol{A}_{7}^{2}$$\varphi_{2,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{2\\}}$(12)$\boldsymbol{A}_{7}^{1}$$\varphi_{1,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{1\\}}$(13)$\boldsymbol{A}_{7}^{3}$$\varphi_{3,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{3\\}}$(14)$\boldsymbol{A}_{7}^{4}$$\varphi_{4,6}$${\boldsymbol{A}_{7}^{6}}^{-\\{4\\}}$(15)
Figure 2: Systems of algebras isolated wrt $\boldsymbol{A}_{7}$

## Summary

In the article, starting from the given algebra, which is the Płonka sum, we
presented a method for determining all its direct systems. Our method consists
in decomposing the Płonka sums into the so-called isolated algebras, for which
we can define the so-called Płonka homomorphisms. Our decomposition can be
performed in two steps (cf. Section 5). The first step consists of three
components. First, we determine the family of algebras isolated wrt a given
algebra. Secondly, we select all frames of isolated algebras. Thirdly, we
modify the obtained frames by replacing the algebras of the given frame with
complement algebras. In the second step, we select a family of
P-homomorphisms. The structure obtained in this way constitutes a direct
system of the initial algebra being a Płonka sum. The presented decomposition
method allows not only to determine all direct systems of a given Płonka sum.
But also check whether a given algebra is a Płonka sum.

In this paper, we focused on algebras with respect to which finitely many
isolated algebras can be determined. An open problem is the question of the
possibility of extending our result to algebras with respect to which we can
determine infinitely many isolated algebras. Additionally, in our work we did
not analyze the problem of uniqueness of the decomposition of the Płonka sum
into isolated algebras, in particular the problem of irreducibility of
algebras that are elements of direct systems of the decomposed sum. Although
we already have partial results regarding the indicated open problems, we will
present their detailed analysis in subsequent works.

## References

  * [1] Bonzio, S., T. Moraschini and M. Pra Baldi, ‘Logics of left variable inclusion and Płonka sums of matrices’, _Archive for Mathematical Logic_ 60:49–76, 2020.
  * [2] Bonzio, S., F. Paoli and M. Pra Baldi, _Logics of Variable Inclusion_ , vol. 59 of Trends in Logic, Springer, 2022.
  * [3] Burris, S. and H. P. Sankappanavar, _A Course in Universal Algebra_ , vol. 78 of Graduate Texts in Mathematics, Springer, 1981.
  * [4] Epstein, R. L., _The semantic foundations of logic. Volume 1: Propositional logics_ , Springer Science+Business Media, Dordrecht, 1990.
  * [5] Graczyńska, E., ‘On normal and regular identities’, _Algebra Universalis_ 27:387–397, 1990.
  * [6] Jarmużek, T. and F. Paoli, ‘Relating logic and relating semantics. History, philosophical applications and some of technical problems’, _Logic and Logical Philosophy_ 30(4):563–577, 2021.
  * [7] Klonowski, M., ‘History of relating logic. The origin and research directions’, _Logic and Logical Philosophy_ 30(4):579–629, 2021.
  * [8] Krajewski, S., ‘One or many logics? (Epstein’s set-assignement semantics for logical calculi)’, _The Journal of Non-Classical Logic_ 8(1):7–33, 1991.
  * [9] Lakser, H., R. Padmanabhan and C. Platt, ‘Subdirect decomposition of Płonka sums’, _Duke Mathematical Journal_ 39:485–488, 1972.
  * [10] Mruczek-Nasieniewska, K., ‘Externally compatible Abelian groups of the type (2,1,0)’, _Logic and Logical Philosophy_ 15:239–250, 2007.
  * [11] Mruczek-Nasieniewska, K., ‘On some extensions of the class of MV-algebras’, _Logic and Logical Philosophy_ 25:35–49, 2016.
  * [12] Mruczek-Nasieniewska, K., ‘On some lattice of varieties related to changes of the type’, in K. Denecke, H.-J.Vogel (eds.), _General Algebra and Applications. Proceedings of the 59th Workshop on General Algebra_ , pp. 147–153, Shaker Verlag, 2000.
  * [13] Mruczek-Nasieniewska, K., ‘P-compatible Abelian groups’, _Logic and Logical Philosophy_ 14:253-263, 2005.
  * [14] Mruczek-Nasieniewska, K., _Równościowe i zdaniowe logiki P-zgodne_ (_Equational and propositional P-compatible logics_), Wydawnictwo UMK, 2013.
  * [15] Mruczek-Nasieniewska, K., ‘Subdirectly irreducible P-compatible Abelian groups’, _Bulletin of the Section of Logic_ 32 (1/2):57–63, 2003.
  * [16] Mruczek-Nasieniewska, K., ‘The varieties defined by P-compatible identities of modular ortholattices’, _Studia Logica_ 95:21–35, 2010.
  * [17] Paoli, F., M. Pra Baldi and D. E. Szmuc, ‘Pure variable inclusion logics’, _Logic and Logical Philosophy_ 30(4):631–652, 2021.
  * [18] Płonka, J., ‘On a method of construction of abstract algebras’, _Fundamenta Mathematicae_ 61(2):183–189, 1967a.
  * [19] Płonka, J., ‘On distributive quasilatticess’, _Fundamenta Mathematicae_ 60:191–200, 1967b.
  * [20] Płonka, J., ‘On equational classes of abstract algebras defined by regular equations’, _Fundamenta Mathematicae_ 64(2):241–247. 1969.
  * [21] Płonka, J., ‘On some decomposition of the sum of a direct system of algebras into a subdirect product’, _Demonstratio Mathematica_ 22(1):235–239, 1989.
  * [22] Płonka, J. and A. Romanowska ‘Semilattice sums’, _Universal Algebra and Quasigroup Theory_ , 123–158, 1992.
  * [23] Płonka, J. and Z. Szylicka, ‘Subdirectly irreducible generalized sums of upper semilattice ordered systems of algebras’, _Algebra Universalis_ 47:201–211, 2002.
  * [24] Romanowska, A. and J. Smith, _Modes_ , World Scientific, 2002.

## Appendix

Examples 26 and 27 present Płonka sums; in turn, Example 28 presents an
algebra that is not a Płonka sum.

######  Example 26.

Let us consider join-semilattice $\mathcal{I}=\langle I,\leqslant\rangle$,
where $I=\\{1,2,3\\}$, wherein $1,2\leqslant 3$, $1\nleqslant 2$ and
$2\nleqslant 1$. Let us consider three two-element Boolean algebras:
$\boldsymbol{B}_{1}=\langle\\{a,b\\};\vee,\wedge,^{\prime}\rangle$,
$\boldsymbol{B}_{2}=\langle\\{c,d\\};\vee,\wedge,^{\prime}\rangle$,
$\boldsymbol{B}_{3}=\langle\\{e,f\\};\vee,\wedge,^{\prime}\rangle$. We define
homomorphisms $\varphi_{1,3}$ and $\varphi_{2,3}$ in the following way:
$\varphi_{1,3}(a)=e$, $\varphi_{1,3}(b)=f$ and $\varphi_{2,3}(c)=e$,
$\varphi_{2,3}(d)=f$.

We can easily check that algebra
$\boldsymbol{A}_{1}=\langle\\{a,b,c,d,e,f\\};\vee,\wedge,^{\prime}\rangle$,
whose operations are defined in Table 3, is the Płonka sum based on direct
system $\langle\mathcal{I},\\{\boldsymbol{B}_{i}\\}_{i\in
I},\allowbreak\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$:

$\wedge$ | $a$ | $b$ | $c$ | $d$ | $e$ | $f$
---|---|---|---|---|---|---
$a$ | $a$ | $a$ | $e$ | $e$ | $e$ | $e$
$b$ | $a$ | $b$ | $e$ | $f$ | $e$ | $f$
$c$ | $e$ | $e$ | $c$ | $c$ | $e$ | $e$
$d$ | $e$ | $f$ | $c$ | $d$ | $e$ | $f$
$e$ | $e$ | $e$ | $e$ | $e$ | $e$ | $e$
$f$ | $e$ | $f$ | $e$ | $f$ | $e$ | $f$
$\vee$ | $a$ | $b$ | $c$ | $d$ | $e$ | $f$
---|---|---|---|---|---|---
$a$ | $a$ | $b$ | $e$ | $f$ | $e$ | $f$
$b$ | $b$ | $b$ | $f$ | $f$ | $f$ | $f$
$c$ | $e$ | $f$ | $c$ | $d$ | $e$ | $f$
$d$ | $f$ | $f$ | $d$ | $d$ | $f$ | $f$
$e$ | $e$ | $f$ | $e$ | $f$ | $e$ | $f$
$f$ | $f$ | $f$ | $f$ | $f$ | $f$ | $f$
$x$ | $x^{\prime}$
---|---
$a$ | $b$
$b$ | $a$
$c$ | $d$
$d$ | $c$
$e$ | $f$
$f$ | $e$

Table 3: Operations of algebra $\boldsymbol{A}_{1}$

######  Example 27.

Let us modify Example 26. Let us consider a join-semilattice
$\mathcal{I}=\langle I,\leqslant\rangle$, where $I=\\{1,2,3\\}$, wherein
$1\leqslant 2\leqslant 3$ and — as in Example 26 — let us consider three two-
element Boolean algebras: $\boldsymbol{B}_{1}$, $\boldsymbol{B}_{2}$,
$\boldsymbol{B}_{3}$. We also consider homomorphisms
$\varphi_{1,2}:B_{1}\longrightarrow B_{2}$ and
$\varphi_{2,3}:B_{2}\longrightarrow B_{3}$ defined in the following way:
$\varphi_{1,2}(a)=c$, $\varphi_{1,2}(b)=d$ and $\varphi_{2,3}(c)=e$,
$\varphi_{2,3}(d)=f$.

Algebra
$\boldsymbol{A}_{2}=\langle\\{a,b,c,d,e,f\\};\vee,\wedge,^{\prime}\rangle$,
whose operations are defined in Table 4, is a Płonka sum based on direct
system $\langle\mathcal{I},\\{\boldsymbol{B}_{i}\\}_{i\in
I},\allowbreak\\{\varphi_{i,j}\\}_{i\leqslant j;i,j\in I}\rangle$:

$\wedge$ | $a$ | $b$ | $c$ | $d$ | $e$ | $f$
---|---|---|---|---|---|---
$a$ | $a$ | $a$ | $c$ | $c$ | $e$ | $e$
$b$ | $a$ | $b$ | $c$ | $c$ | $e$ | $f$
$c$ | $c$ | $c$ | $c$ | $c$ | $e$ | $e$
$d$ | $c$ | $d$ | $c$ | $d$ | $e$ | $f$
$e$ | $e$ | $e$ | $e$ | $e$ | $e$ | $e$
$f$ | $e$ | $f$ | $e$ | $f$ | $e$ | $f$
$\vee$ | $a$ | $b$ | $c$ | $d$ | $e$ | $f$
---|---|---|---|---|---|---
$a$ | $a$ | $b$ | $c$ | $d$ | $e$ | $f$
$b$ | $b$ | $b$ | $d$ | $d$ | $f$ | $f$
$c$ | $c$ | $d$ | $c$ | $d$ | $e$ | $f$
$d$ | $d$ | $d$ | $d$ | $d$ | $f$ | $f$
$e$ | $e$ | $f$ | $e$ | $f$ | $e$ | $f$
$f$ | $f$ | $f$ | $f$ | $f$ | $f$ | $f$
$x$ | $x^{\prime}$
---|---
$a$ | $b$
$b$ | $a$
$c$ | $d$
$d$ | $c$
$e$ | $f$
$f$ | $e$

Table 4: Operations of algebra $\boldsymbol{A}_{2}$

######  Example 28.

Now let the join-semilattice be the structure $\mathcal{I}=\langle
I,\leqslant\rangle$, where $I=\\{1,2\\}$, with $1\leqslant 2$ and — as in
Examples 26 and 27 — two two-element Boolean algebras will be given:
$\boldsymbol{B}_{1}$, $\boldsymbol{B}_{2}$. Consider function
$\varphi_{1,2}:B_{1}\longrightarrow B_{2}$ defined in the following way:
$\varphi_{1,2}(a)=c$, $\varphi_{1,2}(b)=c$.

Algebra
$\boldsymbol{A}_{3}=\langle\\{a,b,c,d,\\};\vee,\wedge,^{\prime}\rangle$, whose
operations are defined in Table 3, is not a Płonka sum.

$\wedge$ | $a$ | $b$ | $c$ | $d$
---|---|---|---|---
$a$ | $a$ | $a$ | $c$ | $c$
$b$ | $a$ | $b$ | $c$ | $c$
$c$ | $c$ | $c$ | $c$ | $c$
$d$ | $c$ | $c$ | $c$ | $d$
$\vee$ | $a$ | $b$ | $c$ | $d$
---|---|---|---|---
$a$ | $a$ | $b$ | $c$ | $d$
$b$ | $b$ | $b$ | $c$ | $d$
$c$ | $c$ | $c$ | $c$ | $d$
$d$ | $d$ | $d$ | $d$ | $d$
$x$ | $x^{\prime}$
---|---
$a$ | $b$
$b$ | $a$
$c$ | $d$
$d$ | $c$

Figure 3: Operations of algebra $\boldsymbol{A}_{3}$

We note that $\varphi_{1,2}$ is not a homomorphism. For example, we have
$\varphi_{1,2}(a^{\prime})\allowbreak=c\neq d=\varphi_{1,2}(a)^{\prime}$. It
is known that in general we have two functions from $\boldsymbol{B}_{1}$ to
$\boldsymbol{B}_{2}$ that could be taken into account to show that
$\boldsymbol{A}_{3}$ is a Płonka sum. The function $\psi\colon
B_{1}\longrightarrow B_{2}$ such that $\psi(a)=c$ and $\psi(b)=d$ is a
homomorphism in the analyzed case, but does not meet the condition of
operation of Płonka sum (see Definition 2). For $\boldsymbol{A}_{3}$ we have
$b\vee c=c$. In turn, the function $\chi\colon B_{1}\longrightarrow B_{2}$
such that $\chi(a)=d$ and $\chi(b)=c$ will not be a homomorphism in the
analyzed case. For $\boldsymbol{A}_{3}$ we have $\chi(a\wedge b)=d\neq
c=\chi(a)\wedge\chi(b)$.

In Example 29 we present isolated algebras specified wrt algebras consider in
the examples presented above.

######  Example 29.

  * •

With respect to algebra $\boldsymbol{A}_{1}$, isolated algebras are
subalgebras of $\boldsymbol{A}_{1}$ with universes: $\\{a,b\\}$, $\\{c,d\\}$
and $\\{a,b,c,d,e,f\\}$. The algebra whose universe is $\\{e,f\\}$ is not
isolated wrt $\boldsymbol{A}_{1}$.

  * •

With respect to algebra $\boldsymbol{A}_{2}$, isolated algebras are
subalgebras of $\boldsymbol{A}_{2}$ with universes: $\\{a,b\\}$,
$\\{a,b,c,d\\}$ and $\\{a,b,c,d,e,f\\}$. Algebras with universes $\\{c,d\\}$
and $\\{e,f\\}$ are not isolated wrt $\boldsymbol{A}_{2}$.

  * •

With respect to the algebra $\boldsymbol{A}_{3}$, isolated algebras are
subalgebras of $\boldsymbol{A}_{3}$ with the universes: $\\{a,b\\}$ and
$\\{a,b,c,d\\}$.

In Example 30 we presented an algebra with respect to which we can determine
finitely many infinite isolated algebras.

######  Example 30.

Let $\mathbb{O}$ be the set of odd natural numbers and $\mathbb{E}$ be the set
of even numbers without zero. Let us consider Płonka sum
$\boldsymbol{A}=\langle\mathbb{N};\ast\rangle$, where $\mathbb{N}$ is the set
of natural numbers without zero and $\ast$ is a binary operation defined in
the following way:

$n\ast m=\begin{cases}1,\quad&\text{if }n,m\in\mathbb{O}\\\
2,\quad&\text{otherwise}.\end{cases}$

Two algebras are isolated wrt $\boldsymbol{A}$:
$\boldsymbol{A}_{1}=\langle\mathbb{O};\ast\rangle$ and $\boldsymbol{A}$. Note
that algebra
$\boldsymbol{A}_{2}:=\langle\mathbb{E};\ast\rangle=\langle\mathbb{N}\setminus\mathbb{O};\ast\rangle$
is not isolated wrt $\boldsymbol{A}$. Function
$\varphi_{1,2}:\mathbb{O}\longrightarrow\mathbb{E}$ s.t.
$\varphi_{1,2}(n)=n+1$ is homomorphism. Płonka sum $\boldsymbol{A}$ can be
determined by direct system
$\langle\langle\\{1,2\\},\leq\rangle,\\{\boldsymbol{A}_{1},\boldsymbol{A}_{2}\\},\\{\varphi_{1,1},\varphi_{1,2},\varphi_{2,2}\\}\rangle$.

Example 31 shows that from a given complement algebra to a complement algebra
of ‘greater’ index not always a homomorphism exists.

######  Example 31.

Let us consider algebra $\boldsymbol{A}_{4}=\langle\\{a,b,c,d\\};\ast\rangle$
where $\ast$ is defined in Table 5.

$\ast$ | $a$ | $b$ | $c$ | $d$
---|---|---|---|---
$a$ | $a$ | $b$ | $c$ | $d$
$b$ | $b$ | $b$ | $c$ | $d$
$c$ | $c$ | $c$ | $d$ | $c$
$d$ | $d$ | $d$ | $d$ | $c$
Table 5: Operations of algebra $\boldsymbol{A}_{4}$

Three algebras are isolated wrt $\boldsymbol{A}_{4}$:
$\boldsymbol{B}_{1}=\langle\\{a\\};\ast\rangle$,
$\boldsymbol{B}_{2}=\langle\\{a,b\\};\ast\rangle$ and
$\boldsymbol{B}_{3}=\langle\\{a,b,c,d\\};\ast\rangle$. There is no
homomorphism from $\boldsymbol{B}_{2}$ to
$\boldsymbol{B}_{3}\setminus\boldsymbol{B}_{2}$. However, algebra
$\boldsymbol{A}_{4}$ is a Płonka sum and can be determined by the following
non-trivial direct system $\langle\langle\\{1,2\\},\\{\langle
1,1\rangle,\langle 2,2\rangle,\langle
1,2\rangle\\}\rangle,\\{\boldsymbol{C}_{1},\allowbreak\boldsymbol{C}_{2}\\},\\{\varphi_{1,1},\varphi_{2,2},\varphi_{1,2}\\}\rangle$,
where $\mathsf{dom}(\boldsymbol{C}_{1})=\\{a\\}$,
$\mathsf{dom}(\boldsymbol{C}_{2})=\\{b,c,d\\}$ and $\varphi_{1,2}(a)=b$.

Example 32 shows that not every homomorphism from complement algebra to a
complement algebra is a homomorphism suitable for determining the Płonka sum
operation in the manner specified in Definition 2.

######  Example 32.

Let us consider Płonka sum
$\boldsymbol{A}_{5}=\langle\\{a,b,c,d\\};\ast\rangle$, where $\ast$ is defined
in Table 6.

$\ast$ | $a$ | $b$ | $c$ | $d$
---|---|---|---|---
$a$ | $a$ | $c$ | $b$ | $a$
$b$ | $c$ | $b$ | $c$ | $c$
$c$ | $b$ | $b$ | $c$ | $b$
$d$ | $a$ | $c$ | $b$ | $d$
Table 6: Operation $\ast$ of algebra $\boldsymbol{A}_{5}$

Płonka sum $\boldsymbol{A}_{5}$ can be determined by direct system
$\langle\langle\\{1,2\\},\\{\langle 1,1\rangle,\langle 1,2\rangle,\langle
2,2\rangle\\}\rangle,\\{\boldsymbol{C}_{1},\boldsymbol{C}_{2}\\},\\{\varphi_{1,1},\varphi_{1,2},\varphi_{2,2}\\}\rangle$,
where $\mathsf{dom}(\boldsymbol{C}_{1})=\\{d\\}$,
$\mathsf{dom}(\boldsymbol{C}_{2})=\\{a,b,c\\}$ and $\varphi_{1,2}(d)=a$.

Three algebras are isolated wrt $\boldsymbol{A}_{5}$:
$\boldsymbol{B}_{1}=\langle\\{d\\};\ast\rangle$,
$\boldsymbol{B}_{2}=\langle\\{a,d\\};\ast\rangle$ and
$\boldsymbol{B}_{3}=\langle\\{a,b,c,\allowbreak d\\};\ast\rangle$. Note that
there is a homomorphism from $\boldsymbol{B}_{2}^{-}$ to
$\boldsymbol{B}_{3}^{-}$ (we put: $\varphi(a)=b$). However, this homomorphism
cannot be taken into account when trying to determine the direct system of
$\boldsymbol{A}_{5}$. We can see that $a\ast c=b\neq c=b\ast c=\varphi(a)\ast
c$. Homomorphism $\varphi$ does not satisfy the condition of Definition 2 for
operation of $\boldsymbol{A}_{5}$.

In Example 33, we present several P-homomorphisms of the algebras discussed in
the examples given above.

######  Example 33.

To indicate example of P-homomorphisms, let us return to algebra
$\boldsymbol{A}_{1}$ from Example 26 and algebra $\boldsymbol{A}_{2}$ from
Example 27.

  * •

We have the following P-homomorphisms wrt $\boldsymbol{A}_{1}$:
$\varphi_{1,3}$ from algebra $\boldsymbol{B}_{1}$ to algebra
$\boldsymbol{B}_{3}$ and $\varphi_{2,3}$ from algebra $\boldsymbol{B}_{2}$ to
algebra $\boldsymbol{B}_{3}$.

  * •

We have the following P-homomorphisms wrt $\boldsymbol{A}_{2}$:
$\varphi_{1,2}$ from algebra $\boldsymbol{B}_{1}$ to algebra
$\boldsymbol{B}_{2}$ and $\varphi_{2,3}$ from algebra $\boldsymbol{B}_{2}$ to
algebra $\boldsymbol{B}_{3}$.

In the case of algebra $\boldsymbol{A}_{5}$ from Example 32, we indicated a
homomorphism that is not a P-homomorphism wrt $\boldsymbol{A}_{5}$.

  * •

Homomorphism $\varphi$ from $\boldsymbol{B}_{2}^{-}$ to
$\boldsymbol{B}_{3}^{-}$ is not a P-homomorphism wrt $\boldsymbol{A}_{5}$. We
noticed that $a\ast c\neq\varphi(a)\ast c$.

Example 34 shows that there might be more than one P-homomorphism from
complement algebras to other complement algebras.

######  Example 34.

Let us consider Płonka sum
$\boldsymbol{A}_{6}=\langle\\{a,b,c,d,e,f\\};\ast\rangle$, where binary
operation $\ast$ is defined in Table 7.

$\ast$ | $a$ | $b$ | $c$ | $d$ | $e$ | $f$
---|---|---|---|---|---|---
$a$ | $a$ | $a$ | $c$ | $c$ | $e$ | $e$
$b$ | $a$ | $a$ | $c$ | $c$ | $e$ | $e$
$c$ | $c$ | $c$ | $c$ | $c$ | $e$ | $e$
$d$ | $c$ | $c$ | $c$ | $c$ | $e$ | $e$
$e$ | $e$ | $e$ | $e$ | $e$ | $e$ | $e$
$f$ | $e$ | $e$ | $e$ | $e$ | $e$ | $e$
Table 7: Operation $\ast$ of algebra $\boldsymbol{A}_{6}$

Three algebras are isolated wrt $\boldsymbol{A}_{6}$:
$\boldsymbol{B}_{1}=\langle\\{a,b\\};\ast\rangle$,
$\boldsymbol{B}_{2}=\langle\\{a,b,c,d\\};\ast\rangle$ and of course
$\boldsymbol{B}_{3}=\boldsymbol{A}_{6}$. Note that there are two homomorphisms
from $\boldsymbol{B}_{1}^{-}=\boldsymbol{B}_{1}$ to $\boldsymbol{B}_{2}^{-}$
and two homomorphisms from $\boldsymbol{B}_{2}$ to $\boldsymbol{B}_{3}^{-}$,
namely:

  * •

$\varphi_{1,2}:B_{1}^{-}\longrightarrow B_{2}^{-}$: $\varphi_{1,2}(a)=c$ and
$\varphi_{1,2}(b)=d$,

  * •

$\psi_{1,2}:B_{1}^{-}\longrightarrow B_{2}^{-}$:
$\psi_{1,2}(a)=\psi_{1,2}(b)=c$,

  * •

$\varphi_{2,3}:B_{2}^{-}\longrightarrow B_{3}^{-}$: $\varphi_{2,3}(c)=e$ and
$\varphi_{2,3}(d)=f$,

  * •

$\psi_{2,3}:B_{2}^{-}\longrightarrow B_{3}^{-}$:
$\psi_{2,3}(c)=\psi_{2,3}(d)=e$.

It is easy to check that each of these functions is a P-homomorphism. So, we
have one isolated algebras frame and four possible choices of P-homomorphisms.