# Surface sensitivity of the spin Seebeck effect

A. Aqeel  I. J. Vera-Marun  B. J. van Wees  T. T. M. Palstra
<EMAIL_ADDRESS>Zernike Institute for Advanced Materials, University of
Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands

###### Abstract

We have investigated the influence of the interface quality on the spin
Seebeck effect (SSE) of the bilayer system yttrium iron garnet (YIG) $-$
platinum (Pt). The magnitude and shape of the SSE is strongly influenced by
mechanical treatment of the YIG single crystal surface. We observe that the
saturation magnetic field ($H_{\text{sat}}^{\text{SSE}}$) for the SSE signal
increases from 55.3 mT to 72.8 mT with mechanical treatment. The change in the
magnitude of $H_{\text{sat}}^{\text{SSE}}$ can be attributed to the presence
of a perpendicular magnetic anisotropy due to the treatment induced surface
strain or shape anisotropy in the Pt/YIG system. Our results show that the SSE
is a powerful tool to investigate magnetic anisotropy at the interface.

##  I Introduction

The discovery of the spin Seebeck effect (SSE) Uchida  _et al._ (2010a) in
insulators triggered the modern era of the field of spin caloritronics Bauer
_et al._ (2012). In insulators, instead of moving charges, only spin
excitations (magnons) drive the non-equilibrium spin currents. In the spin
Seebeck effect, spin currents are thermally excited in a ferromagnet FM and
detected in a normal metal NM deposited on the FM. The bilayer NM/FM system in
the SSE provides the opportunity to separately tune the properties of both
layers to optimize the magnitude and magnetic field dependence of the SSE
effect. The platinum (Pt) and yttrium iron garnet (YIG) bilayer system has
attracted considerable attention for studying the spin Seebeck effect Uchida
_et al._ (2010a, b); Kehlberger  _et al._ (2013); Kirihara  _et al._ (2012)
and for other spin dependent transport experiments Kajiwara  _et al._ (2010);
Burrowes  _et al._ (2012); Uchida  _et al._ (2010a); Sandweg  _et al._ (2011);
Heinrich  _et al._ (2011); Rezende  _et al._ (2013); Qiu  _et al._ (2012);
Vlietstra  _et al._ (2013); Flipse  _et al._ (2014). Platinum (Pt) has a large
inverse spin Hall response Ishida  _et al._ (2013) whereas YIG is an ideal
ferromagnetic insulator due to low magnetic damping Bauer  _et al._ (2012);
Kajiwara  _et al._ (2010); Chikazumi (1977) and a large band gap Jia  _et al._
(2011) at room temperature.

The origin of the spin Seebeck effect is commonly explained by the difference
in the magnon temperature in the FM and the phonon temperature in the NM,
$\Delta T_{mp}$ Xiao  _et al._ (2010); Adachi  _et al._ (2013). When the
temperature gradient $\nabla T$ is applied across the NM/FM system, it creates
a $\Delta T_{mp}$ based on the thermal conductivities of the magnon and the
phonon subsystems Xiao  _et al._ (2010). This $\Delta T_{mp}$ induces a spin
current density at the interface which is detected in the normal metal Pt by
the inverse spin Hall effect (ISHE). The ISHE signal depends on a scaling
parameter, the interfacial SSE coefficient $L_{s}$, related to how efficiently
the spin current density can be created across the interface under a certain
$\Delta T_{mp}$. The resulting spin Seebeck signal scales linearly with the
length of the NM ($l_{Pt}$), therefore for the Pt/YIG system

$V_{\text{ISHE}}\propto l_{Pt}\ .\ L_{s}\ .\ \nabla T$ (1)

The scaling parameter $L_{s}$ is proportional to the real part of the spin
mixing conductance $g_{r}^{\uparrow\downarrow}$ at the interface. The spin
mixing conductance $g_{r}^{\uparrow\downarrow}$ and therefore the SSE are very
sensitive to the interface quality Jungfleisch  _et al._ (2013). In recent
years substantial effort has been made to improve the spin mixing conductance
on thin films of YIG Qiu  _et al._ (2013); Jungfleisch  _et al._ (2013) and
bulk crystals Jia  _et al._ (2011); Uchida  _et al._ (2013). Unlike thin
films, bulk crystals need an extra surface polishing step for the device
fabrication, due to the initial surface roughness. The polishing of the
crystal surface can influence the spin mixing conductance in several ways.
Apart from changing the surface roughness, mechanical polishing can change the
magnetic structures at the interface by inducing a small perpendicular
anisotropy at the surface layer of the YIG crystal Craik and Griffiths (1959);
Craik and Tebble (1961); Coleman (1974). However, the effect of polishing on
the spin Seebeck effect (SSE) has not yet been systematically studied. In this
paper, we report the effect of mechanical surface treatment of the YIG single
crystals on the SSE effect. This systematic study reveals the surface
sensitivity of the SSE and indicates new ways of surface modification for
improved thermoelectric efficiency.

##  II Experimental technique

In this study, we use the longitudinal configuration Uchida  _et al._ (2010b)
for the spin Seebeck effect where the temperature gradient is applied across a
NM/FM interface and parallel to the spin current direction $J_{s}$. In Fig.
1(a), we illustrate schematically the device configuration for measuring the
SSE used in this study. The sample consists of a single crystal YIG slab and a
Pt film sputtered on a (111) surface of the YIG crystal. When an out-of-plane
(along z-axis) temperature gradient is applied to the Pt/YIG stack, spin waves
are thermally excited. The spin waves inject a spin current along the z-axis
and polarize the spins in the Pt film close to the interface parallel to the
magnetization of the YIG crystal, as illustrated in Fig. 1(b). Due to the
strong spin-orbit coupling in the Pt-film, the spin polarization $\sigma$ is
converted to an electrical voltage $V_{\text{ISHE}}$. The single crystals of
YIG with the same purity were used in all measurements. The YIG crystals were
grown by the floating zone method along the (111) direction and commercially
available from Crystal Systems Corporation company, Hokuto, Yamanashi Japan .
A diamond saw was used to cut the crystals. The YIG crystals were cleaned
ultrasonically first in acetone and then ethanol baths.

Figure 1:  (a) Device configuration of the longitudinal SSE where $\nabla T$
represents the temperature gradient across the Pt/YIG system. (b) Detection of
spin current by the ISHE. The orange arrows indicate the spin polarization
$\sigma$ at the interface of the Pt/YIG system. M, $J_{S}$ and
$E_{\text{ISHE}}$ represent the magnetization of YIG, spatial direction of the
thermally generated spin current, and electric field induced by the ISHE,
respectively. $\theta$ represents the angle between the external magnetic
field H in the x-y plane and the x axis. (c) AFM height image of a single
crystal YIG surface (20 x 20 $\mu m^{2}$) for sample S1. (d) a comparison
between the magnetic field dependence of $V_{\text{ISHE}}$ at $\Delta T$ = 3.6
K for sample S1 and the magnetization M of the YIG crystal.

Three different types of surfaces were prepared for samples S1 - S3 by the
following treatments:

  * •

For S1: the YIG crystals were grinded with abrasive grinding papers (SiC P1200
- SiC P4000) at 150 rpm for 1h. After grinding, diamond particles were used
with a sequence of 9 $\mu m$, 3 $\mu m$ and 1 $\mu m$ at 300 rpm for 30 mins,
respectively. To remove the surface strain or surface damage due to diamond
particles Craik and Griffiths (1959); Craik and Tebble (1961); Coleman (1974),
colloidal silica OPS (oxide polishing suspension) with a particle size of 40
nm was used, which can give mechanical as well as chemical polishing. To
remove the residuals of polishing particles, samples were heated at 200 ∘C for
1h at ambient conditions. Then crystals were cleaned by acetone and ethanol in
an ultrasonics bath before depositing the Pt layer on top.

  * •

For S2: grinding, polishing and cleaning of the samples were done in the same
way as described for S1. However, the colloidal silica OPS was not used for
sample S2. Thus, the strained or damaged surface layer due to diamond
polishing was retained.

  * •

For S3: no mechanical polishing was done to obtain flat surfaces as done for
samples S1 and S2. After cleaning in the same way as done for samples S1 and
S2, Pt was deposited on the unpolished YIG crystal surface.

Table 1: Surface treatment, surface roughness, and orientation of the YIG crystals for different samples. Samples | Polishing | Roughness | Orientation
---|---|---|---
S1 | Silica |  $<$ 3 nm | (111)
S2 | diamond |  $\geq$ 12 nm | (111)
S3 | no |  $>$ 300 nm | (111)

The surface treatments are summarized in Table 1. The measurements of the spin
Seebeck effect (SSE) were performed in the following way. The samples were
magnetized in the xy plane of the YIG crystal by an external magnetic field H,
as shown in Fig. 1. To excite the spin waves an external heater generates a
temperature gradient $\nabla T$ across the Pt/YIG stack where the temperature
of heat sink is denoted as T. The thickness of the YIG (Pt bar) is 3 mm (15
nm) for all samples. Regarding the lateral dimension of the Pt bar, the length
(width) varies from 5 mm-3 mm (2.5-1.5) with all samples having ratios 2:1.
The surface of the YIG crystals was analyzed by atomic force microscopy (AFM)
before deposition of the Pt film on top. The observed spin Seebeck signals
show a small offset which we removed. The field at which 95$\%$ of the SSE
signal saturates is defined as $H_{\text{sat}}^{\text{SSE}}$. The
magnetization M of the YIG crystal with a dimension of 2 mm x 1 mm was
measured with a SQUID magnetometer.

##  III Results and discussion

Fig. 1(c) shows the AFM height image of sample S1 with a surface roughness
smaller than 3 nm. A distinct $V_{\text{ISHE}}$ signal appears and saturates
around $\sim$ 55.3 mT, which is close to the field required to saturate the
magnetization of the YIG crystal, as illustrated in Fig. 1(d). Similarly, the
YIG surface of sample S2 was analyzed by AFM. Fig. 2(a) shows that sample S2
has a surface roughness around $\sim$ 12 nm with strip-like trenches at the
surface. A clear spin Seebeck response has been observed for sample S2 by
changing the applied magnetic field H. The signal saturates at relatively
higher values of H ($\sim$ 66.1 mT) compared to the magnetization of YIG as
shown in Fig. 2b.

Figure 2:  (a) AFM height image of a single crystal YIG surface for sample S2
(20 x 20 $\mu m^{2}$). (b) Comparison between the H dependence of
$V_{\text{ISHE}}$ at $\Delta T$ = 3.6 K in sample S2 and the magnetization M
of the YIG crystal. (c) Temperature dependence of
$H_{\text{sat}}^{\text{SSE}}$. The inset shows $H_{\text{sat}}^{\text{SSE}}$
as a function of $T^{\varepsilon}$ where $\varepsilon$ = 2. (d)
$V_{\text{ISHE}}$ as a function of the external magnetic field direction
$\theta$ in the Pt/YIG system at a fixed magnetic field 80 mT.

In addition, we checked the magnetic field dependence of the spin Seebeck
response at low-temperatures for sample S2, the temperature dependence of the
$H_{\text{sat}}^{\text{SSE}}$ is given in Fig. 2(c). As the YIG crystal is a
3D isotropic ferrimagnet, the temperature dependence of the magnetic order
parameter obeys a $T^{2}$ universality scaling Solt Jr. (1962). To understand
the temperature dependence of $H_{\text{sat}}^{\text{SSE}}$, we fitted
$H_{\text{sat}}^{\text{SSE}}$ at low temperatures by assuming $T_{c}$ = 553 K
as shown in the inset of Fig. 2(c). The temperature dependence of
$H_{\text{sat}}^{\text{SSE}}$ closely obeys the $T^{2}$ universality behavior
of the order parameter of the YIG crystal with exponent $\varepsilon$ = 2. It
suggests that the $H_{\text{sat}}^{\text{SSE}}$ directly depends on the order
parameter of the YIG crystal. To confirm further the origin of the observed
signal, H is rotated in the x-y plane. The $V_{\text{ISHE}}$ signal follows
the expected sinusoidal dependence for a spin Seebeck signal, as shown in Fig.
2(d).

Unlike the samples S1 and S2, sample S3 has a very large surface roughness
($>$ 300 nm) as shown in Fig. 3(a). Nevertheless, a clear spin Seebeck signal
was observed as shown in Fig 3(b).

Figure 3:  (a) AFM height image of the YIG surface for sample S3 (20 x 20 $\mu
m^{2}$) and (b) a comparison between the H dependence of $V_{\text{ISHE}}$ at
$\Delta$T = 7.5 K in sample S3 and the magnetization M of the YIG crystal.

From equation 1, it follows that the inverse spin Hall voltage
$V_{\text{{ISHE}}}$ is proportional to the applied temperature gradient
$\nabla T$ and the length of the Pt bar $l_{Pt}$. $V_{\text{ISHE}}$ increases
by reducing the thickness of the Pt film $t_{Pt}$, for both the spin pumping
Castel  _et al._ (2012) and the SSE Saiga  _et al._ (2014) experiments.
Therefore to compare samples with different Pt thickness we can define a
parameter C as follows Castel  _et al._ (2012); Saiga  _et al._ (2014);
Nakayama  _et al._ (2012):

$C=\frac{1}{\textrm{tanh}[\dfrac{t_{Pt}}{2\lambda_{Pt}}]\ \rho_{Pt}\
\dfrac{l_{Pt}}{t_{Pt}}}\ \frac{V_{\text{ISHE}}}{\nabla T}$ (2)

Here, $\nabla T$ is defined as the temperature difference across the Pt/YIG
stack normalized with the thickness of the YIG crystal, $\rho_{Pt}$ is the
resistivity of Pt and $\lambda_{Pt}$ is the spin diffusion length of Pt. In
these experiments, unlike $\rho_{Pt}$, $\lambda_{Pt}$ cannot be measured
directly therefore we assumed that it remains constant for different samples.
Note that for all samples discussed here $t_{Pt}$ $>$ 2 $\lambda_{Pt}$ (where
$\lambda_{Pt}$ = $1.5~{}nm$ Vlietstra  _et al._ , 2013; Flipse  _et al._ ,
2014) so the tanh$[\dfrac{t_{Pt}}{2\lambda_{Pt}}]$ term is approximately equal
to 1 leading to $V_{\text{{ISHE}}}$ $\propto$ $1/t_{Pt}$. Moreover, the C
parameter is independent of the YIG thickness when the thickness is larger
than the magnon mean free path and therefore it can be used as an indicator of
changes in other parameters related to the interfacial mechanisms of the SSE.

Table 2: Comparison of the resistance R of the Pt film, the C parameter and the $H_{\text{sat}}^{\text{SSE}}$ for the SSE response in bulk single crystals and thin films |  | Bulk crystals |  |  | Thin films
---|---|---|---|---|---
| S1 | S2 | S3 |  Ref. Uchida  _et al._ (2010b) |  Ref. Flipse  _et al._ (2014)
R ($\Omega$) | 33.8 | 52.2 | 119 | - | -
C (10-8V $\Omega^{-1}$ K-1) | 0.917 | 1.369 | 0.043 | 0.554 | 1.105
$H_{\text{sat}}^{\text{SSE}}$ (mT) | 55.3 | 66.1 | 72.8 | 40 | 2.5

The resistance of the Pt film varies for the samples S1-S3, nevertheless all
samples have similar resistance within an order of magnitude as shown in Table
2. The observed change in the resistance is correlated with the roughness of
the crystals, although we do not observe the same scaling for the SSE
response. For example, the resistance of sample S2 is 50$\%$ higher than
sample S1 whereas the SSE signal for sample S2 is only 30$\%$ higher than
sample S1. Furthermore the resistance of sample S3 is almost four times bigger
than sample S1, however the SSE response actually follows the opposite trend,
it is actually more than an order of magnitude lower than the response of the
samples S1 and S2. Therefore, we establish that the dominant mechanism
relevant for the observed differences in the SSE signal is not the resistivity
of the NM films but the quality of the NM/FM interface. Sample S1 gives a C
parameter that is comparable to the value reported for thin films and bulk
crystals as shown in Table 2. However, sample S2 shows 30$\%$ bigger and
sample S3 shows more than an order of magnitude smaller value of the C
parameter than sample S1. The observed variation in the value of the C
parameter indicates the importance of mechanical treatment induced surface
effects that we will discuss below.

Based on the experimental conditions listed in Table 1 and the results
summarised in Table 2, we propose a possible mechanism for our observations.
Fig. 4(a-c) schematically illustrates possible interface morphologies and the
surface magnetization for the NM/FM system, for different interface conditions
between the NM film and the FM crystal. Fig. 4(a) represents the case for a NM
film deposited on an atomically flat FM crystal. Here, the case for sample S1
corresponds to Fig. 4(a). Fig. 4(b) depicts a situation for a NM deposited on
a flat FM crystal but having a small perpendicular anisotropy at the surface.
The situation represented in Fig. 4(b) corresponds to the case for sample S2.
The surface of the YIG crystal for sample S2 contains trenches due to
polishing of the YIG crystal with coarse diamond particles as shown in Fig.
2(a). The trenches at the interface can induce strain or shape anisotropy
resulting in a perpendicular anisotropy at the interface. The presence of a
small perpendicular anisotropy at the interface would increase the
$H_{\text{sat}}^{\text{SSE}}$ compared to the bulk magnetization of the YIG
crystal, which has been clearly observed for sample S2 (see Fig. 2(b)).

In addition, the magnitude of the SSE signal can also change if the mechanical
polishing changes the atomic termination for the density of Fe atoms that are
in direct contact with the Pt metal. If the density of Fe atoms at the surface
is larger than the bulk of the YIG, the observed SSE signal would be larger
Uchida  _et al._ (2013); Jia  _et al._ (2011). The increase in the SSE signal
for sample S2 compared to sample S1 can be attributed to different chemical
termination due to polishing with coarse diamond particles. Fig. 4(c) shows
the case for a rough interface between the NM and the FM crystal which
corresponds to the situation for sample S3. In case of sample S3, the lack of
further mechanical treatment after cutting with a diamond saw leaves a very
rough surface of the YIG crystal. The $H_{\text{sat}}^{\text{SSE}}$ is around
72.8 mT for sample S3 as shown in Fig. 3(b). The increase in the value of
$H_{\text{sat}}^{\text{SSE}}$ for sample S3 compared to the magnetization of
YIG can be due to a non-uniform magnetization at the interface resulting from
high surface roughness of the YIG crystal.

Figure 4:  (a-c) A schematic illustration of the interface morphologies of the
NM/FM system for different surface treatments of the FM where orange arrows
represent $\nabla T$: (a) An atomically flat interface, (b) an interface with
a perpendicular anisotropy and (c) a rough interface. (d) Comparison between
the magnitude of the C parameter and (e) comparison between the line profile
of the SSE signal as a function of H for all samples.

Fig. 4(d) gives a comparison of the magnitude of the SSE signal in terms of
the C parameter (as defined in equation 2) for samples with different
mechanical treatments. The observed signal for sample S3 is smallest compared
to other samples. This can be explained due to the increase of surface
roughness Burrowes  _et al._ (2012); Jia  _et al._ (2011) resulting in the
small spin mixing conductance at the interface. The sample S1 has the lowest
surface roughness, however the SSE signal observed for sample S2 is the
largest compared to the samples S1 and S3 as shown in Fig. 4(d). Therefore,
for the largest roughness of sample S3 we see a relation between roughness and
the SSE signal, but not for the samples S1 and S2. Hence, the roughness is not
the only parameter and this might be related to the more abrasive nature of
the diamond particles leaving a different chemical termination at the
interface.

To compare the line profile of the $V_{\text{ISHE}}$ signal, in Fig. 4(e) the
signals are normalized by their value at H = 90 mT, where they reach
saturation. Fig. 4(e) shows that the line profile of the SSE signal changes
with moving from soft silica to coarse diamond particle polishing. For the
samples S1 and S2 the $V_{\text{ISHE}}$ is very small at zero applied field
compared to the value measured at 90 mT. However, for sample S3 the
$V_{\text{ISHE}}$ is almost 64$\%$ of the value measured at H = 90 mT. The
value of $H_{\text{sat}}^{\text{SSE}}$ is highest for sample S3 with the
largest surface roughness and lowest for sample S1 with the smallest surface
roughness. Therefore, the $H_{\text{sat}}^{\text{SSE}}$ directly correlates
with the roughness of sample. The large deviation in the magnitude of SSE
signal and the $H_{\text{sat}}^{\text{SSE}}$ in the YIG crystals with
different surface treatments emphasizes the surface sensitivity of the spin
Seebeck effect. Our results indicate that not only the surface roughness but
actual atomic structures and chemical termination at the interface also play
an important role in the SSE.

##  IV Conclusions

In conclusion, we have shown a strong dependence of the spin Seebeck signal on
the interface condition of the Pt/YIG bilayer system. We observed a change of
18 mT in the saturation field of the SSE signal by changing the type of
polishing. Furthermore we observe the change in the magnitude of the SSE
signal for different samples. No definite relation has been found between the
SSE response and the sample roughness. However, we observe a direct
correlation between the $H_{\text{sat}}^{\text{SSE}}$ and the roughness of
sample, as the former increases by moving from soft toward coarse particle
polishing. To understand the origin of the magnitude and change in the
saturation field $H_{\text{sat}}^{\text{SSE}}$ for the observed SSE signal,
due to different types of surface treatments, the crystal surfaces need to be
investigated further in detail.

###### Acknowledgements.

We would like to acknowledge J. Baas, H. Bonder, G.H. ten Brink, M. de Roosz
and J. G. Holstein for technical assistance. This work is supported by the
Foundation for Fundamental Research on Matter (FOM), the Netherlands
Organisation for Scientific Research (NWO), Marie Curie ITN Spinicur NanoLab
NL, and the Zernike Institute for Advanced Materials National Research
Combination.

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On left legal semigroups111Mathematics Subject Classification: 20M10, 20M12
Keywords: Semigroup, free semigroup, variety of semigroups

Attila Nagy222email<EMAIL_ADDRESS>
Department of Algebra

Budapest University of Technology and Economics

Műegyetem rkp. 3, Budapest, 1111, Hungary

Abstract

In this paper we study semigroups satisfying the identity $aba=ab$.

##  1 Introduction

Many important combinatorial structures such as real and complex hyperplane
arrangements, interval greedoids, matroids and oriented matroids have the
structure of a left regular band (see, for example, [1, 9, 12, 23]). A
semigroup $S$ is called a left regular band if every element of $S$ is an
idempotent element, and $S$ satisfies the identity $aba=ab$. Left regular
bands are examined by many authors. See, for example, papers [2, 3, 5, 9, 23,
24, 26, 28] and the book [18]. In [21, 22], the authors examined semigroups
satisfying the identiy $aba=ab$. Using the terminology of [22], such
semigroups are called left legal semigroups. The class $\mathcal{LLS}$ of all
left legal semigroups is a variety. Therefore, it is natural to find free
semigroups and subdirectly irreducible semigroups in this variety. In Section
3, we construct free left legal semigroups $F_{\mathcal{LLS}}(X)$ for
arbitrary nonempty sets $X$. The subdirectly irreducible left legal semigroups
are examined in Section 4. We show that a semigroup containing at least two
elements is a subdirectly irreducible left legal semigroup satisfying the
identity $ab=a^{2}b$ if and only if it is either a subdirectly irreducible
left regular band or a two-element zero semigroup. We also show that a
semigroup containing at least two elements is a subdirectly irreducible (left
legal) semigroup satisfying the identity $ab=ac$ if and only if it is either a
two-element left zero semigroup or a two-element zero semigroup. The concept
of the retract ideal extension of semigroups is used effectively in many cases
in the study of the structure of semigroups. In Section 4, we give a necessary
and sufficient condition for a left legal semigroup to be a retract ideal
extension of a left regular band by a zero semigroup. One of the basic
concepts of semigroup theory is the semilattice decomposition. In Section 5,
this concept is in the centre of investigations. We show that a semigroup $S$
is a semilattice indecomposable left legal semigroup if and only if $S^{2}$ is
a left zero semigroup. The left (resp., right, weakly) separative semigroups
are examined in many papers. In Section 6, these semigroups are the focus. We
show that a left legal semigroup is right separative if and only if it is
weakly separative. It is also proved that every left legal left separative
semigroup is commutative. In Section 7, the lattice of all left legal
semigroup varieties is examined.

##  2 Preliminaries

By a _semigroup_ we mean a multiplicative semigroup, i.e. a nonempty set
together with an associative multiplication. A nonempty subset $I$ of a
semigroup $S$ is said to be an _ideal_ of $S$ if $sa,as\in I$ for every $a\in
I$ and $s\in S$. If $I$ is an ideal of a semigroup $S$, then the relation
$\varrho_{I}$ on $S$ defined by $(a,b)\in\varrho_{I}$ if and only if $a=b$ or
$a,b\in I$ is a congruence on $S$, which is called the _Rees congruence on $S$
determined by $I$_. The equivalence classes of $S$ mod $\varrho_{I}$ are $I$
itself and every one-element set $\\{a\\}$ with $a\in S\setminus I$. The
factor semigroup $S/\varrho_{I}$ is called the _Rees factor semigroup of $S$
modulo $I$_. We shall write $S/I$ instead of $S/\varrho_{I}$. We may describe
$S/I$ as the result of collapsing $I$ into a single (zero) element, while the
elements of $S$ outside of $I$ retain their identity. A semigroup $S$
containing a zero element $0$ is called a _zero semigroup_ if $ab=0$ for every
$a,b\in S$. It is obvious that if $S$ is a semigroup, then $S^{2}$ is an ideal
of $S$, and the Rees factor semigroup $S/S^{2}$ is a zero semigroup. Let $B$
and $Q$ be disjoint semigroups, $Q$ having a zero element. A semigroup $S$ is
called an _ideal extension_ of $B$ by $Q$ if $S$ contains $B$ as an ideal, and
if the Rees factor semigroup $S/B$ is isomorphic with $Q$. An ideal $I$ of a
semigroup $S$ is called a _retract ideal_ if there is a homomorphism of $S$
onto $I$ which leaves the elements of $I$ fixed. Such a homomorphism is called
a _retract homomorphism_ of $S$ onto $I$. In this case we say that $S$ is a
_retract ideal extension_ of $I$.

We say that a semigroup $S$ is a _subdirect product_ of semigroups $S_{i}$
($i\in I$) if $S$ is isomorphic to a subsemigroup $T$ of the direct product of
semigroups $S_{i}$ ($i\in I$) such that the restriction of the projection
homomorphisms to $T$ are surjective. A semigroup $S$ is said to be
_subdirectly irreducible_ if whenever $S$ is written as a subdirect product of
a family of semigroups $S_{i}$ ($i\in I$), then, for at least one $j\in I$,
the projection homomorphism $\pi_{j}$ maps $S$ onto $S_{j}$ isomorphically.
Subdirect decompositions of a semigroup $S$ are closely connected with
congruences on $S$ (see, for example, [18, I.3.6]). If $\alpha_{i}$ ($i\in I$)
are congruences on a semigroup $S$ and $\cap_{i\in I}\alpha_{i}=\iota_{S}$,
the equality relation on $S$, then $S$ is a subdirect product of the factor
semigroups $S/\alpha_{i}$. Conversely, if a semigroup is a subdirect product
of semigroups $S_{i}$ ($i\in I$) and $\alpha_{i}$ is the congruence on $S$
induced by the projection homomorphism $\pi_{i}$ ($i\in I$), then $\cap_{i\in
I}\alpha_{i}=\iota_{S}$.

If $\cal C$ is a class of semigroups, then a congruence $\varrho$ on a
semigroup $S$ is called a $\cal C$-_congruence_ if the factor semigroup
$S/\varrho$ belongs to $\cal C$. An element $e$ of a semigroup is called an
_idempotent element_ if $e^{2}=e$. A semigroup in which every element is an
idempotent element is called a _band_. A commutative band is called a
_semilattice_. It is clear that the universal relation on an arbitrary
semigroup is a semilattice congruence. A semigroup $S$ is said to be
_semilattice indecomposable_ if the universal relation is the only semilattice
congruence on $S$. By Theorem of [25], every semigroup $S$ has a least
semilattice congruence $\eta_{S}$ whose classes are semilattice indecomposable
semigroups. In other words, every semigroup is a semilattice of semilattice
indecomposable semigroups. A band satisfying the identity $aba=ab$ is called a
_left regular band_. A semigroup satisfying the identity $ab=a$ is called a
_left zero semigroup_. By the dual of [18, II.3.12. Proposition], a band is
left regular if and only if it is a semilattice of left zero semigroups. It is
clear that every left zero semigroup is left legal and semilattice
indecomposable.

A semigroup $S$ is called a left legal semigroup if it satisfies the identity
$aba=ab$. Every left regular band is a left legal semigroup. The converse is
also true for semigroups $S$ containing a right identity element, because
$a^{2}=aea=ae=a$ is satisfied for every element $a$ of $S$ and a right
identity element $e$ of $S$.

A semigroup $S$ is called an _$\cal L$ -commutative semigroup_ [15] if, for
every elements $a,b\in S$, there is an element $x\in S^{1}$ such that
$ab=xba$. A semigroup $S$ is called a _right weakly commutative semigroup_ if,
for every $a,b\in S$, there exists $x\in S$ and a positive integer $n$ such
that $(ab)^{n}=xa$. It is clear that every $\cal L$-commutative semigroup is
right weakly commutative.

A semigroup $S$ is said to be a _right separative semigroup_ if $ab=b^{2}$ and
$ba=a^{2}$ imply $a=b$ for every $a,b\in S$. A _left separative semigroup_ is
defined analogously. A semigroup $S$ is called a _weakly separative semigroup_
if $a^{2}=ab=b^{2}$ implies $a=b$ for every $a,b\in S$.

For an arbitrary semigroup $S$,

$S^{1}=\begin{cases}S,&\textit{if $S$ has an identity element};\\\ S\cup
1,&\textit{otherwise},\end{cases}$

where $S\cup 1$ is the semigroup which is obtained by the adjunction of an
identity element $1$ to $S$.

For notions and notations not defined but used in this paper, we refer the
reader to [6].

##  3 Free left legal semigroups

Let $\mathcal{V}$ be a non-trivial variety of algebras, i.e. $\mathcal{V}$
contains algebras with more than one element. An algebra
$F_{\mathcal{V}}(X)\in\mathcal{V}$ is said to be a free $\mathcal{V}$-algebra
on the set $X$ if $X$ generates $F_{\mathcal{V}}(X)$, and every mapping of $X$
into any algebra $A$ from $\mathcal{V}$ can be extended to a homomorphism of
$F_{\mathcal{V}}(X)$ into $A$. In this section we construct free left legal
semigroups $F_{\mathcal{LLS}}(X)$ for arbitrary nonempty sets $X$.

First we prove two lemmas which will be used throughout the paper.

######  Lemma 1.

If $S$ is a left legal semigroup, then every element of $S^{2}$ is an
idempotent element. Moreover, $a^{k}=a^{2}$ for every $a\in S$ and every
integer $k\geq 2$.

###### Proof.

Let $S$ be a left legal semigroup. For every elements $a$ and $b$ of $S$, we
have

$(ab)^{2}=a(bab)=aba=ab.$

Thus every element of $S^{2}$ is an idempotent element. For every $a\in S$,
$a^{3}=a^{2}$ by definition, and hence $a^{k}=a^{2}$ for every integer $k\geq
2$. ∎

######  Lemma 2.

For arbitrary elements $a$ and $b$ of a left legal semigroup, $ab=ab^{2}$.

###### Proof.

Using Lemma 1, we get $ab=(ab)^{2}=(aba)b=ab^{2}$. ∎

Let $X$ be a nonempty set, whose elements are also said to be letters. A
finite sequence of letters from $X$ is said to be a word. It is known [26]
that the free left regular band on $X$ consists of all words of the letters
from $X$ which have no duplicate letters, and the multiplication $\cdot$ is
defined by
$\omega_{1}\cdot\omega_{2}=\omega_{1}\exists_{\omega_{1}}(\omega_{2})$, where
$\exists_{\omega_{1}}(\omega_{2})$ means that we remove from $\omega_{2}$ all
letters contained in $\omega_{1}$. As we will see below, the elements of the
free left legal semigroup differ only in a small detail from the elements of
the free left regular band. Before constructing the free left legal semigroup,
we note that Lemma 1 and Lemma 2 imply that every product of elements of a
left legal semigroup can be reduced to a product of the form
$a^{k}_{1}a_{2}\cdots a_{n}$, equal to the original product, in which the
elements $a_{1}\dots a_{n}$ are pairwise distinct and $k=1$ or $k=2$.
Accordingly, for a nonempty set $X$, let $F_{\mathcal{LLS}}(X)$ denote the set
of all nonempty words of the letters from $X$ in which each letter $x\in X$
can occur at most once or twice, in the latter case in the form $xx$ at the
beginning of the word. For an arbitrary word $\omega\in F_{\mathcal{LLS}}(X)$,
let $\omega^{*}=\omega$, if the letters of $\omega$ are pairwise distinct, and
let $\omega^{*}$ be the word which can be obtained from $\omega$ by deleting
one of its first two letters if they are equal to each other. We define an
operation $\circ$ on $F_{\mathcal{LLS}}(X)$: for arbitrary words
$\omega_{1}=x_{1}\dots x_{n}$ and $\omega_{2}=y_{1}\dots y_{m}$ of
$F_{\mathcal{LLS}}(X)$, let

$\omega_{1}\circ\omega_{2}=\begin{cases}xx\exists_{\omega_{1}}(\omega^{*}_{2}),&\textit{if
$\omega_{1}=x=y_{1}$ for some $x\in X$,}\\\
\omega_{1}\exists_{\omega_{1}}(\omega^{*}_{2}),&\textit{otherwise.}\end{cases}$
(1)

######  Example 1.

If $X=\\{x,y\\}$, then $F_{\mathcal{LLS}}(X)=\\{x,xx,y,yy,xy,xxy,yx,yyx\\}$.
The operation $\circ$ on $F_{\mathcal{LLS}}(X)$ is given by Table 1.

$\circ$ | $x$ | $xx$ | $y$ | $yy$ | $xy$ | $xxy$ | $yx$ | $yyx$
---|---|---|---|---|---|---|---|---
$x$ | $xx$ | $xx$ | $xy$ | $xy$ | $xxy$ | $xxy$ | $xy$ | $xy$
$xx$ | $xx$ | $xx$ | $xxy$ | $xxy$ | $xxy$ | $xxy$ | $xxy$ | $xxy$
$y$ | $yx$ | $yx$ | $yy$ | $yy$ | $yx$ | $yx$ | $yyx$ | $yyx$
$yy$ | $yyx$ | $yyx$ | $yy$ | $yy$ | $yyx$ | $yyx$ | $yyx$ | $yyx$
$xy$ | $xy$ | $xy$ | $xy$ | $xy$ | $xy$ | $xy$ | $xy$ | $xy$
$xxy$ | $xxy$ | $xxy$ | $xxy$ | $xxy$ | $xxy$ | $xxy$ | $xxy$ | $xxy$
$yx$ | $yx$ | $yx$ | $yx$ | $yx$ | $yx$ | $yx$ | $yx$ | $yx$
$yyx$ | $yyx$ | $yyx$ | $yyx$ | $yyx$ | $yyx$ | $yyx$ | $yyx$ | $yyx$
Table 1:

######  Theorem 3.

If $X$ is an arbitrary nonempty set, then $(F_{\mathcal{LLS}}(X);\circ)$ is
the free left legal semigroup on the set $X$.

###### Proof.

Let $X$ be a nonempty set and $F_{X}$ be the free semigroup on $X$, i.e. the
set of all arbitrary nonempty words of letters from $X$. Let $I$ denote the
set of identities $uvu=uv$, $u^{3}=u^{2}$, and $uv^{2}=uv$. By Lemma 1 and
Lemma 2, a semigroup is a left legal semigroup if and only if it satisfies the
identities belonging to $I$. By [8, Section V.], the free left legal semigroup
is obtained as a quotient semigroup $F_{X}/\theta_{I}$, where the congruence
$\theta_{I}$ is constructed on $F_{X}$ by defining two words to be
$\theta_{I}$-congruent if they are equal or we can transform one into the
other by a finite sequence of $I$-transformations, i.e. applications of the
identities $uvu=uv$, $u^{3}=u^{2}$, and $uv^{2}=uv$. The factor semigroup
$F_{X}/\theta_{I}$ belongs to the variety $\mathcal{LLS}$ of all left legal
semigroups and freely generated in $\mathcal{LLS}$ by the elements $[x]$
($x\in X$), where $[x]$ denotes the $\theta_{I}$-class of $F_{X}$ containing
the letter $x$. We show that the groupoid $(F_{\mathcal{LLS}}(X);\circ)$ is an
isomorphic image of the factor semigroup $F_{X}/\theta_{I}$, from which the
statement of the theorem follows. Let $(F_{X},\rightarrow)$ be a length
reducing rewrite system where $\rightarrow$ sends a word $\omega\in F_{X}$ to
a shorter word of $F_{X}$ which can be obtained from $\omega$ by applying one
of the identities in $I$. For example, $\rightarrow$ send the word $xxxyz$ to
the word $xxyz$ by applying the identity $u^{3}=u^{2}$. A word $\omega\in
F_{X}$ is said to be a normal form if there is no $\hat{\omega}\in F_{X}$ with
$\omega\rightarrow\hat{\omega}$. Let $\xrightarrow{*}$ is the reflexive-
transitive closure of $\rightarrow$, that is
$\omega\xrightarrow{*}\tilde{\omega}$ means that $\omega=\tilde{\omega}$ or
there are words $\omega_{1},\dots,\omega_{n}\in F_{X}$ such that
$\omega=\omega_{1}\rightarrow\omega_{2}\rightarrow\dots\rightarrow\omega_{n}=\tilde{\omega}$.
A word $\omega^{\prime}\in F_{X}$ is said to be a normal form of a word
$\omega\in F_{X}$ if $\omega^{\prime}$ is a normal form and
$\omega\xrightarrow{*}\omega^{\prime}$. For a word $\omega\in F_{X}$, let
$v(\omega)=z_{1}z_{2}\dots z_{k}$ be the word of $F_{X}$ which satisfies the
following conditions: $\\{z_{1},z_{2},\dots,z_{k}\\}$ is the set of all
letters from $X$ belonging to $\omega$, and $z_{i-1}$ appears in $\omega$
sooner than $z_{i}$ ($i=2,\dots,k$). Obviously, $z_{1}$ equals the first
letter of $\omega$ for arbitrary $\omega\in F_{X}$. If the first two letters
of a word $\omega\in F_{X}$ of length at least two are not equal, then $z_{2}$
equals the second letter of $\omega$. It is easy to see that every word
$\omega=x_{1}x_{2}\dots x_{n}$ has one and only one normal form, denoted by
$\omega^{\prime}$: if $n=1$ then $\omega^{\prime}=\omega$; if $n\geq 2$, then
$\omega^{\prime}$ can be obtained by the help of $v(\omega)=z_{1}z_{2}\dots
z_{k}$ as follows

$\omega^{\prime}=\begin{cases}x_{1}z_{2}\dots z_{k},&\textit{if $x_{1}\neq
x_{2}$}\\\ x_{1}x_{1}z_{2}\dots z_{k},&\textit{if $x_{1}=x_{2}$.}\end{cases}$
(2)

From this result we get that the set of all normal forms of $F_{X}$ equals
$F_{\mathcal{LLS}}(X)$. From (2) it follows that
$\omega^{\prime}_{1}=\omega^{\prime}_{2}$ is satisfied for words
$\omega_{1}=x_{1}x_{2}\dots x_{n}$ and $\omega_{2}=y_{1}y_{2}\dots y_{m}$
($n,m\geq 2$) of $F_{X}$ if and only if the prefixes $x_{1}x_{2}$ and
$y_{1}y_{2}$ are equal, and $v(\omega_{1})=v(\omega_{2})$. It is also true
that words $\omega_{1}=x_{1}x_{2}\dots x_{n}$ and $\omega_{2}=y_{1}y_{2}\dots
y_{m}$ ($n,m\geq 2$) are $\theta_{I}$-congruent if and only if both of
$x_{1}x_{2}=y_{1}y_{2}$ and $v(\omega_{1})=v(\omega_{2})$ are satisfied. Thus
every $\theta_{I}$-class of $F_{X}$ contains one and only one normal form,
also considering that $[x]=\\{x\\}$ for every $x\in X$. Since
$F_{\mathcal{LLS}}(X)$ is the set of all normal forms,
$\Phi:[\omega]\to\omega^{\prime}$ is a bijective mapping of the factor set
$F_{X}/\theta_{I}$ onto the set $F_{\mathcal{LLS}}(X)$, where
$\omega^{\prime}$ denotes the normal form of $\omega$. It is easy to see that,
for arbitrary words $\omega_{1}$ and $\omega_{2}=y_{1}\dots y_{m}$ of $F_{X}$,

$(\omega_{1}\omega_{2})^{\prime}=\begin{cases}xx\omega^{\prime}_{1}\exists_{\omega^{\prime}_{1}}((\omega^{\prime}_{2})^{*}),&\textit{if
$\omega_{1}=x=y_{1}$ for some $x\in X$,}\\\
\omega^{\prime}_{1}\exists_{\omega^{\prime}_{1}}((\omega^{\prime}_{2})^{*}),&\textit{otherwise.}\end{cases}$
(3)

Using (1) and (3), we have
$(\omega_{1}\omega_{2})^{\prime}=\omega^{\prime}_{1}\circ\omega^{\prime}_{2}$.
Then

$\Phi([\omega_{1}]\bullet[\omega_{2}])=\Phi([\omega_{1}\omega_{2}])=(\omega_{1}\omega_{2})^{\prime}=\omega^{\prime}_{1}\circ\omega^{\prime}_{2}=\Phi([\omega_{1}])\circ\Phi([\omega_{2}]).$

Thus $\Phi$ is a homomorphism. Consequently $\Phi$ is an isomorphism of the
semigroup $F_{X}/\theta_{I}$ onto the groupoid $(F_{\mathcal{LLS}}(X);\circ)$.
Hence $(F_{\mathcal{LLS}}(X);\circ)$ is a semigroup, which is isomorphic to
the free left legal semigroup $(F_{X}/\theta_{I};\bullet)$. The set $X$ is the
free generating system of $(F_{\mathcal{LLS}}(X);\circ)$. ∎

##  4 Retract ideal extensions of left regular bands by zero semigroups

By Lemma 1, if $S$ is a left legal semigroup, then every element of $S^{2}$ is
an idempotent element. Thus every left legal semigroup is an ideal extension
of a left regular band by a zero semigroup. In this section we focus on the
case when this extension is retract. We note that if a semigroup $S$ is an
ideal extension of a band $B$ by a zero semigroup, then $B=S^{2}$, because
$B\subseteq S^{2}$ and the assumption that the Rees factor semigroup $S/B$ is
a zero semigroup implies $S^{2}\subseteq B$.

######  Theorem 4.

The following three conditions on a semigroup $S$ are equivalent.

  * (i)

$S$ is a retract ideal extension of a left regular band by a zero semigroup.

  * (ii)

$S$ is a subdirect product of a left regular band and a zero semigroup.

  * (iii)

$S$ is a left legal semigroup satisfying the identity $ab=a^{2}b$.

###### Proof.

$(i)\Rightarrow(ii)$: Assume that $S$ is a retract ideal extension of a left
regular band $B$ by a zero semigroup. Then $B=S^{2}$ is a retract ideal of
$S$. Let $\varphi$ be a retract homomorphism of $S$ onto $B$. It is easy to
see that

$ker_{\varphi}\cap\varrho_{B}=\iota_{S},$

where $ker_{\varphi}$ denotes the kernel of $\varphi$, and $\varrho_{B}$
denotes the Rees congruence on $S$ modulo the ideal $B$. Then, by [18, I.3.6],
$S$ is a subdirect product of a left regular band and a zero semigroup,
because the factor semigroup $S/ker_{\varphi}$ is isomorphic to the left
regular band $B$, and the Rees factor semigroup $S/B$ is a zero semigroup.

$(ii)\Rightarrow(iii)$: It is obvious, because every subsemigroup of the
direct product of a left regular band and a zero semigroup is a left legal
semigroup satisfying the identity $ab=a^{2}b$.

$(iii)\Rightarrow(i)$: Assume that $S$ is a left legal semigroup satisfying
the identity $ab=a^{2}b$. By Lemma 1, $S$ is an ideal extension of the left
regular band $S^{2}$ by the zero semigroup $S/S^{2}$. Let $\varphi$ be the
mapping of $S$ into $S^{2}$ defined by the following way: for an arbitrary
$a\in S$, $\varphi(a)=a^{2}$. As $S^{2}$ is a band, $\varphi$ maps $S$ onto
$S^{2}$ and leaves the elements of $S^{2}$ fixed. Let $a,b\in S$ be arbitrary
elements. Using Lemma 1, Lemma 2 and the assumption that $S$ satisfies the
identity $ab=a^{2}b$, we have

$\varphi(a)\varphi(b)=a^{2}b^{2}=a^{2}b=ab=(ab)^{2}=\varphi(ab).$

Then $\varphi$ is a retract homomorphism of $S$ onto $S^{2}$. Consequently $S$
is a retract ideal extension of the left regular band $S^{2}$ by the zero
semigroup $S/S^{2}$. ∎

######  Example 2.

On the set $S=\\{x,e,f,g,h\\}$, consider the operation defined by Table 2.

| $x$ | $e$ | $f$ | $g$ | $h$
---|---|---|---|---|---
$x$ | $e$ | $e$ | $e$ | $g$ | $h$
$e$ | $e$ | $e$ | $e$ | $g$ | $h$
$f$ | $f$ | $f$ | $f$ | $g$ | $h$
$g$ | $g$ | $g$ | $g$ | $g$ | $g$
$h$ | $h$ | $h$ | $h$ | $h$ | $h$
Table 2:

It is a matter of checking to see that this operation is associative.
$S^{2}=\\{e,f,g,h\\}$ is a semilattice of the left zero semigroups
$S_{1}=\\{e,f\\}$ and $S_{2}=\\{g,h\\}$, and hence $S^{2}$ is a left regular
band. The mapping $\varphi:s\mapsto s^{2}$ is a retract homomorphism of $S$
onto $S^{2}$. Then, by Theorem 4, $S$ is a subdirect product of the left
regular band $S^{2}$ and the two-element zero semigroup $S/S^{2}=\\{x,0\\}$.

In the free left legal semigroup $F_{\mathcal{LLS}}(X)$ considered in Example
1, $xy\neq xxy$. Thus not every left legal semigroup satisfies the conditions
in Theorem 4.

######  Theorem 5.

A semigroup containing at least two elements is a subdirectly irreducible left
legal semigroup satisfying the identity $ab=a^{2}b$ if and only if it is
either a subdirectly irreducible left regular band or a two-element zero
semigroup.

###### Proof.

A left regular band is a left legal semigroup and satisfies the identity
$ab=a^{2}b$. It is obvious that a two-element zero semigroup is a subdirectly
irreducible left legal semigroup satisfying the identity $ab=a^{2}b$. To prove
the converse assertion, assume that $S$ is a subdirectly irreducible left
legal semigroup which has at least two elements and satisfies the identity
$ab=a^{2}b$. Then, by $(ii)$ of Theorem 4, $S$ is either a subdirectly
irreducible left regular band or a subdirectly irreducible zero semigroup $Z$.
Since a zero semigroup is subdirectly irreducible if and only if it has at
most two elements, we have $\mid Z\mid=2$. ∎

For subdirectly irreducible left regular bands, we refer the reader to [28,
Theorem 2.16].

######  Theorem 6.

The following three conditions on a semigroup $S$ are equivalent.

  * (i)

$S$ is a retract ideal extension of a left zero semigroup by a zero semigroup.

  * (ii)

$S$ is a subdirect product of a left zero semigroup and a zero semigroup.

  * (iii)

$S$ satisfies the identity $ab=ac$ (and so $S$ is a left legal semigroup).

###### Proof.

$(i)\Rightarrow(ii)$: Assume that $S$ is a retract ideal extension of a left
zero semigroup $L$ by a zero semigroup. Then $L$ is an ideal of $S$, and the
Rees factor semigroup $S/L$ is a zero semigroup. Let $\varphi$ be a retract
homomorphism of $S$ onto $L$. Then

$ker_{\varphi}\cap\varrho_{L}=\iota_{S},$

and hence [18, I.3.6] implies that $S$ is a subdirect product of the left zero
semigroup $L\cong S/ker_{\varphi}$ and the zero semigroup $S/L$.

$(ii)\Rightarrow(iii)$: It is obvious, because every subsemigroup of the
direct product of a left zero semigroup and a zero semigroup satisfies the
identity $ab=ac$.

$(iii)\Rightarrow(i)$: Let $S$ be a semigroup satisfying the identity $ab=ac$.
Then $S$ is a left legal semigroup. By Lemma 1, $S^{2}$ is a band. Then, for
arbitrary elements $e,f\in S^{2}$, we have

$ef=e^{2}=e,$

using the assumption that $S$ satisfies the identity $ab=ac$. Hence $S^{2}$ is
a left zero semigroup. Let $\varphi$ be the mapping of $S$ into $S^{2}$
defined by the following way: for an arbitrary $a\in S$, $\varphi(a)=a^{2}$.
As $S^{2}$ is a band, $\varphi$ maps $S$ onto $S^{2}$ which leaves the
elements of $S^{2}$ fixed. For arbitrary $a,b\in S$,

$\varphi(a)\varphi(b)=a^{2}b^{2}=a(abb)=a(bab)=(ab)^{2}=\varphi(ab),$

using again the assumption that $S$ satisfies the identity $ax=ay$. Thus
$\varphi$ is a homomorphism. Consequently $S$ is a retract ideal extension of
the left zero semigroup $S^{2}$ by the zero semigroup $S/S^{2}$. ∎

######  Example 3.

On the set $S=\\{x,y,b,c\\}$, consider the operation defined by Table 3.

| $x$ | $y$ | $b$ | $c$
---|---|---|---|---
$x$ | $b$ | $b$ | $b$ | $b$
$y$ | $c$ | $c$ | $c$ | $c$
$b$ | $b$ | $b$ | $b$ | $b$
$c$ | $c$ | $c$ | $c$ | $c$
Table 3:

It is easy to check that $S$ is a semigroup with respect to this operation,
and $S$ is an ideal extension of the left zero semigroup $S^{2}=\\{b,c\\}$ by
the zero semigroup $S/S^{2}=\\{x,y,0\\}$. Moreover, the mapping $\varphi\colon
s\mapsto s^{2}$ is a retract homomorphism of $S$ onto $S^{2}$. Thus the
semigroup $S$ satisfies the conditions of Theorem 6.

The next example shows that if a semigroup is an ideal extension of a left
zero semigroup by a zero semigroup, then it does not necessarily satisfy the
conditions of Theorem 6.

######  Example 4.

On the set $S=\\{a,b,c,d,e,f\\}$, consider the operation defined by Table 4.

| $a$ | $b$ | $c$ | $d$ | $e$ | $f$
---|---|---|---|---|---|---
$a$ | $e$ | $c$ | $e$ | $c$ | $e$ | $c$
$b$ | $d$ | $f$ | $d$ | $f$ | $d$ | $f$
$c$ | $c$ | $c$ | $c$ | $c$ | $c$ | $c$
$d$ | $d$ | $d$ | $d$ | $d$ | $d$ | $d$
$e$ | $e$ | $e$ | $e$ | $e$ | $e$ | $e$
$f$ | $f$ | $f$ | $f$ | $f$ | $f$ | $f$
Table 4:

It is easy to see that $S$ is a semigroup which is an ideal extension of the
left zero semigroup $S^{2}=\\{c,d,e,f\\}$ by the zero semigroup
$S/S^{2}=\\{a,b,0\\}$. Since $ab\neq ac$, the semigroup $S$ does not satisfy
the conditions of Theorem 6.

######  Theorem 7.

A semigroup containing at least two elements is a subdirectly irreducible
semigroup satisfying the identity $ab=ac$ if and only if it is either a two-
element left zero semigroup or a two-element zero semigroup.

###### Proof.

It is obvious that every two-element semigroup is subdirectly irreducible.
Moreover, every left zero semigroup and every zero semigroup satisfies the
identity $ab=ac$. To prove the converse assertion, let $S$ be a subdirectly
irreducible semigroup which has at least two elements and satisfies the
identity $ab=ac$. Then, by $(ii)$ of Theorem 6, $S$ is either a left zero
semigroup or a zero semigroup. Since a left zero semigroup and, similarly, a
zero semigroup is subdirectly irreducible if and only if it contains at most
two elements, $S$ is either a two-element left zero semigroup or a two-element
zero semigroup. ∎

##  5 The least semilattice congruence on left legal semigroups

In the study of the structure of semigroups, the semilattice decomposition of
semigroups plays a basic role. It is proved in [25] that every semigroup $S$
has a least semilattice congruence $\eta_{S}$ whose classes are semilattice
indecomposable semigroups; the induced partition of $S$ is called a
semilattice decomposition of $S$, and the congruence classes of $\eta_{S}$ are
said to be the semilattice components of $S$.

A semigroup $S$ is said to be an _archimedean semigroup_ if, for arbitrary
$a,b\in S$, there is a positive integer $n$ such that $a^{n}\in SbS$ and
$b^{n}\in SaS$. Every archimedean semigroup is semilattice indecomposable. It
is proved in [20] that a semigroup is a semilattice of archimedean semigroups
if and only if it is a Putcha semigroup. Recall that a semigroup $S$ is called
a _Putcha semigroup_ if, for every $a,b\in S$, the assumption $a\in
S^{1}bS^{1}$ implies $a^{m}\in S^{1}b^{2}S^{1}$ for some positive integer $m$.

######  Theorem 8.

Every left legal semigroup is a Putcha semigroup.

###### Proof.

Let $S$ be a left legal semigroup. It is clear that $S$ is $\cal
L$-commutative, and hence right weakly commutative. By [16, Theorem 4.1] and
[16, Lemma 2.2], every right weakly commutative semigroup is a Putcha
semigroup, which proves our assertion. ∎

######  Remark 1.

Theorem 8 and [16, Theorem 2.1] together imply that if $S$ is a left legal
semigroup, then,

$\eta_{S}=\\{(a,b)\in S\times S:\ a^{n}\in SbS,\ b^{n}\in SaS\ \hbox{for a
positive integer}\ n\\},$

where $\eta_{S}$ denotes the least semilattice congruence on $S$. This implies
that a left legal semigroup is semilattice indecomposable if and only if it is
archimedean.

For an arbitrary semigroup $S$, let $\tau_{S}$ denote the binary relations on
$S$ defined by the following way: $(a,b)\in\tau_{S}$ for elements $a,b\in S$,
if and only if there is a positive integer $n$ such that $a^{n}b=a^{n+1}$ and
$b^{n}a=b^{n+1}$. By [16, Lemma 4.1], $\tau_{S}$ and its dual $\sigma_{S}$ are
equivalence relations on an arbitrary semigroup $S$. Equivalence relations
$\tau_{S}$ and $\sigma_{S}$ are studied by many authors in special classes of
semigroups. In the book [6], Theorem 4.14 asserts that if $S$ is a commutative
semigroup, then $\tau_{S}=\sigma_{S}$ is a least weakly separative congruence
on $S$. In [7, Theorem 5.4], Chrislock shown that if $S$ is a medial
semigroup, then $\tau_{S}$ is the least left separative congruence on $S$, and
$\sigma_{S}$ is the least right separative congruence on $S$. In [13, Theorem
6], Mukherjee proved that $\sigma_{S}$ is a weakly separative congruence on a
quasi-commutative semigroup $S$. It is also conjectured that $\sigma_{S}$ is
the least weakly separative congruence. Pondelicek proved [19, Theorem 5] that
if $S$ is a duo semigroup, then $\tau_{S}\cap\sigma_{S}$ is the least weakly
separative congruence on $S$. Since every quasicommutative semigroup is a duo
semigroup, this result implies that $\tau_{S}\cap\sigma_{S}$ is the least
weakly separative congruence on a quasi-commutative semigroup, giving a
negative answer for Mukherjee’s conjecture. Generalizing and completing the
above results, in [14, Theorem 1] it is proved that $\tau_{S}\cap\sigma_{S}$
is the least weakly separative congruence on a weakly commutative semigroup
$S$. Using this result, in [17] the weakly separative weakly commutative
semigroups are characterized.

Since every left legal semigroup is right weakly commutative, [16, Theorem
4.7] implies that $\tau_{S}$ is a weakly separative congruence on a left legal
semigroup. The next theorem gives an additional result about $\tau_{S}$ if $S$
is a left legal semigroup.

######  Theorem 9.

If $S$ is a left legal semigroup, then $\tau_{S}=\eta_{S}$, where $\eta_{S}$
is the least semilattice congruence on $S$.

###### Proof.

We use Remark 1. First we show that $\eta_{S}\subseteq\tau_{S}$. Assume
$(a,b)\in\eta_{S}$ for elements $a,b\in S$. Then, by Remark 1, $a^{n}=xby$ and
$b^{n}=uav$ for a positive integer $n$ and for some elements $x,y,u,v\in S$.
We can suppose $n\geq 2$. Then

$a^{n}b=(xby)b=x(byb)=xby=a^{n}=a^{n+1},$

because $a^{k}=a^{2}$ for every integer $k\geq 2$ by Lemma 1. Similarly,
$b^{n}a=b^{n+1}$. Thus $(a,b)\in\tau_{S}$. Hence $\eta_{S}\subseteq\tau_{S}$.
To show $\tau_{S}\subseteq\eta_{S}$, assume $(a,b)\in\tau_{S}$ for elements
$a,b\in S$. Then $a^{n}b=a^{n+1}$ and $b^{n}a=b^{n+1}$ for some positive
integer $n\geq 2$. Thus

$a^{n+2}=a^{n}ba\in SbS\quad\hbox{and}\quad b^{n+2}=b^{n}ab\in SaS.$

Hence $(a,b)\in\eta_{S}$. Thus $\tau_{S}\subseteq\eta_{S}$. Consequently
$\eta_{S}=\tau_{S}$. ∎

######  Remark 2.

By Lemma 1, if $a$ is an arbitrary element of a left legal semigroup, then
$a^{k}=a^{2}$ is satisfied for every integer $k\geq 2$. Thus
$(a,b)\in\tau_{S}$ is satisfied for elements $a$ and $b$ of a left legal
semigroup if and only if $a^{2}b=a^{2}$ and $b^{2}a=b^{2}$.

######  Theorem 10.

The following three conditions on a semigroup $S$ are equivalent.

  * (i)

$S$ is a semilattice indecomposable left legal semigroup.

  * (ii)

$S$ is a left legal semigroup satisfying the identity $a^{2}b=a^{2}$.

  * (iii)

$S^{2}$ is a left zero semigroup.

###### Proof.

By Theorem 9 and Remark 2, a left legal semigroup $S$ is semilattice
indecomposable if and only if $\tau_{S}$ is the identity relation on $S$, i.e.
$a^{2}b=a^{2}$ is satisfied for every elements $a$ and $b$ of $S$. Thus it is
sufficient to show that $(ii)$ and $(iii)$ are equivalent.

$(ii)\Rightarrow(iii)$: Assume that $S$ is a left legal semigroup satisfying
the identity $a^{2}b=a^{2}$. Since $S$ is left legal, $S^{2}$ is a band by
Lemma 1. Let $e,f\in S^{2}$ be arbitrary elements. Then

$ef=e^{2}f=e^{2}=e,$

i.e. $S^{2}$ is a left zero semigroup.

$(iii)\Rightarrow(ii)$: Assume that $S^{2}$ is a left zero semigroup. Let
$a,b\in S$ be arbitrary elements. Then

$aba=(ab)(ab)a=(ab)(aba)=ab.$

Thus $S$ is a left legal semigroup. Moreover,

$a^{2}b=(a^{2}a^{2})b=a^{2}(a^{2}b)=a^{2}.$

∎

The semigroup $S$ defined in Example 3 is a semilattice indecomposable left
legal semigroup in which $S^{2}$ is a left zero semigroup such that it is a
retract ideal of $S$. The semigroup $S$ defined in Example 4 is also a
semilattice indecomposable left legal semigroup in which $S^{2}$ is a left
zero semigroup, but $S^{2}$ is not a retract ideal of $S$.

By [18, II. 1.4. Lemma] and the dual of [18, II. 3. 12. Proposition], a band
$B$ is a left regular band if and only if the classes of its least semilattice
congruence $\eta_{B}$ are left zero semigroups. Moreover, $(e,f)\in\eta_{B}$
is satisfied for elements $e,f\in B$ if and only if $ef=e$ and $fe=f$. The
next theorem shows how to get the classes of the least semilattice congruence
on a left legal semigroup.

######  Theorem 11.

Let $S$ be a left legal semigroup and $L_{i}$ ($i\in I$) be left zero
subsemigroups of $S$ such that the left regular band $S^{2}$ is a semilattice
of $L_{i}$ ($i\in I$). Then the classes of the least semilattice congruence
$\eta_{S}$ on $S$ are $S_{i}=\\{a\in S:\ a^{2}\in L_{i}\\}$ ($i\in I$).

###### Proof.

Let $a,b\in S$ be arbitrary elements. By Theorem 9 and Remark 2,
$(a,b)\in\eta_{S}$ if and only if $(a,b)\in\tau_{S}$, i.e.

$a^{2}b=a^{2}\quad\hbox{and}\quad b^{2}a=b^{2}.$

By Lemma 2, it is equivalent to the condition

$a^{2}b^{2}=a^{2}\quad\hbox{and}\quad b^{2}a^{2}=b^{2},$

i.e. $a^{2},b^{2}\in L_{i}$ for some $i\in I$, which is equivalent to the
condition that $a,b\in S_{i}$. Thus the classes of the least semilattice
congruence $\eta_{S}$ on $S$ are $S_{i}$ ($i\in I$). ∎

From Theorem 11 it follows that the maximal semilattice image of a left legal
semigroup $S$ is equal to that of its maximal sub-left regular band $S^{2}$,
which in turn is isomorphic to the semilattice of its $\mathcal{L}$-classes =
$\mathcal{J}$-classes.

It is easy to see that $\\{xx\\}$, $\\{yy\\}$, and $\\{xy,xxy,yx,yyx\\}$ are
left zero subsemigroups of the free left legal semigroup
$F_{\mathcal{LLS}}(X)$ considered in Example 1 such that
$F^{2}_{X}=\\{xx,yy,xy,xxy,yx,yyx\\}$ is a semilattice of them. By Theorem 11,
the classes of the least semilattice congruence on $F_{\mathcal{LLS}}(X)$ are
$A=\\{x,xx\\}$, $B=\\{y,yy\\}$, and $C=\\{xy,xxy,yx,yyx\\}$. It can also be
directly proved that $A$, $B$, and $C$ are semilattice indecomposable
subsemigroups of $F_{\mathcal{LLS}}(X)$, and the equivalence relation on
$F_{\mathcal{LLS}}(X)$ whose classes are $A$, $B$, and $C$ is a semilattice
congruence on $F_{\mathcal{LLS}}(X)$.

##  6 Right and left separativity of left legal semigroups

######  Theorem 12.

The following two conditions on a left legal semigroup $S$ are equivalent.

  * (i)

$S$ is right separative.

  * (ii)

$S$ is weakly separative.

###### Proof.

$(i)\Rightarrow(ii)$: Let $S$ be a right separative left legal semigroup. To
show that $S$ is weakly separative, assume $a^{2}=ab=b^{2}$ for elements
$a,b\in S$. Using Lemma 1, $ab=b^{2}$ implies

$bab=b^{3}=b^{2},$

and hence

$ba=bab=b^{2}=a^{2}.$

Then

$ab=b^{2}\quad\hbox{and}\quad ba=a^{2}.$

Since $S$ is right separative, we get $a=b$. Consequently $S$ is weakly
separative.

$(ii)\Rightarrow(i)$: Let $S$ be a weakly separative left legal semigroup. To
show that $S$ is right separative, assume $ab=b^{2}$ and $ba=a^{2}$ for
elements $a,b\in S$. Using the first equation and Lemma 1, we get

$ba=bab=b^{3}=b^{2},$

and hence

$b^{2}=ba=a^{2}.$

Since $S$ is weakly separative, we get $a=b$. Consequently $S$ is right
separative. ∎

######  Remark 3.

It is clear that every left zero semigroup containing at least two elements is
left legal and weakly separative, but not left separative. Thus a weakly
separative left legal semigroup is not left separative, in general.

######  Theorem 13.

Every left separative left legal semigroup is commutative.

###### Proof.

Let $S$ be a left separative left legal semigroup. Then, for every $a,b\in S$,

$(ab)(ba)=(aba)(bab)=(ab)^{3}=(ab)^{2},$

using Lemma 1. Similarly,

$(ba)(ab)=(ba)^{2}.$

Since $S$ is left separative, we get $ab=ba$, and hence $S$ is commutative. ∎

##  7 On the lattice of all left legal semigroup varieties

The class of all varieties of semigroups forms a lattice under the following
operations: for varieties $\mathcal{X}$ and $\mathcal{Y}$, their join
${\mathcal{X}}\vee{\mathcal{Y}}$ is the variety generated by the set-
theoretical union of ${\mathcal{X}}$ and $\mathcal{Y}$ (as classes of
semigroups), and their meet $\mathcal{X}\wedge\mathcal{Y}$ coincides with the
set-theoretical intersection of $\mathcal{X}$ and $\mathcal{Y}$. A variety
$\mathcal{V}$ of semigroups is said to be a left legal semigroup variety, if
every semigroup belonging to $\mathcal{V}$ is left legal. The set of all left
legal semigroup varieties form a sublattice of the lattice of all semigroup
varieties. This sublattice denoted by ${\mathcal{L}}({\bf LLS})$. In this
section we use the following notations:

  * $\mathcal{A}$: the variety of semigroups satisfying the identities $aba=ab$, $ab=a^{2}b$;

  * $\mathcal{B}$: the variety of semigroups satisfying the identity $ab=ac$;

  * $\mathcal{C}$: the variety of semigroups satisfying the identities $abc=acb$, $ab=a^{2}b$;

  * $\mathcal{D}$: the variety of semigroups satisfying the identities $ab=ba$, $ab=a^{2}b$;

  * $\mathcal{LRB}$: the variety of all left regular bans ($a^{2}=a$, $aba=ab$);

  * $\mathcal{LNB}$: the variety of all left normal bands ($a^{2}=a$ $abc=acb$);

  * $\mathcal{SL}$: the variety of all semilattices ($a^{2}=a$, $ab=ba$);

  * $\mathcal{LZ}$: the variety of all left zero semigroups ($ab=a$);

  * $\mathcal{ZM}$: the variety of all zero semigroups ($ab=0$);

  * $\mathcal{T}$: the trivial variety ($a=b$).

We note that if a semigroup $S$ belongs to the variety $\mathcal{C}$, then
$aba=a^{2}b=ab$ for arbitrary $a,b\in S$, and hence $S$ is a left legal
semigroup. Then, by Lemma 2, $ab=ab^{2}$ is also satisfied for arbitrary
$a,b\in S$. Thus a semigroup belongs to the variety $\mathcal{C}$ if and only
if it satisfies the identities $abc=acb$, $ab=a^{2}b$, and $ab=ab^{2}$.

It is easy to see that

$\mathcal{LZ},\mathcal{ZM}\subseteq\mathcal{B}\subseteq\mathcal{A},$
$\mathcal{SL}\subseteq\mathcal{LNB}\subseteq{LRB}\subseteq\mathcal{A},$
$\mathcal{D}\subseteq\mathcal{C}\subseteq\mathcal{A}.$

Since $\mathcal{A}\subseteq\mathcal{LLS}$, all of the above varieties are in
the lattice of all left legal semigroup varieties.

It is known (see, for example, [8] and [18]) that

$\mathcal{B}=\mathcal{ZM}\vee\mathcal{LZ},$
$\mathcal{C}=\mathcal{B}\vee\mathcal{SL}=\mathcal{B}\vee\mathcal{LNB}=\mathcal{B}\vee\mathcal{D}=\mathcal{D}\vee\mathcal{LZ}=\mathcal{LNB}\vee\mathcal{ZM},$
$\mathcal{D}=\mathcal{ZM}\vee\mathcal{SL},$
$\mathcal{LNB}=\mathcal{LZ}\vee\mathcal{SL},$
$\mathcal{B}\wedge\mathcal{LNB}=\mathcal{LZ},\
\mathcal{C}\wedge\mathcal{LRB}=\mathcal{LNB},\
\mathcal{D}\wedge\mathcal{LNB}=\mathcal{SL}.$

######  Theorem 14.

For the varieties $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $\mathcal{D}$,
$\mathcal{LRB}$, and $\mathcal{ZM}$, the following equations hold:
$\mathcal{A}=\mathcal{ZM}\vee\mathcal{LRB}=\mathcal{D}\vee\mathcal{LRB}=\mathcal{C}\vee\mathcal{LRB}=\mathcal{B}\vee\mathcal{LRB}$.

###### Proof.

$\mathcal{A}=\mathcal{ZM}\vee\mathcal{LRB}$: Since the varieties
$\mathcal{ZM}$ and $\mathcal{LRB}$ are subvarieties of the variety
$\mathcal{A}$, we have $\mathcal{ZM}\vee\mathcal{LRB}\subseteq\mathcal{A}$. By
Theorem 4, every semigroup belonging to $\mathcal{A}$ is a subdirect product
of a left regular band and a zero semigroup, and hence it belongs to
$\mathcal{ZM}\vee\mathcal{LRB}$. Thus
$\mathcal{A}\subseteq\mathcal{ZM}\vee\mathcal{LRB}$. Consequently
$\mathcal{A}=\mathcal{ZM}\vee\mathcal{LRB}$.

$\mathcal{A}=\mathcal{D}\vee\mathcal{LRB}$: Using equations
$\mathcal{D}=\mathcal{ZM}\vee\mathcal{SL}$,
$\mathcal{A}=\mathcal{ZM}\vee\mathcal{LRB}$, and the fact that
$\mathcal{LS}\subseteq\mathcal{LRB}$, we have

$\mathcal{D}\vee\mathcal{LRB}=\mathcal{ZM}\vee\mathcal{SL}\vee\mathcal{LRB}=\mathcal{ZM}\vee\mathcal{LRB}=\mathcal{A}.$

$\mathcal{A}=\mathcal{C}\vee\mathcal{LRB}$: Using equations
$\mathcal{C}=\mathcal{B}\vee\mathcal{D}$,
$\mathcal{B}=\mathcal{ZM}\vee\mathcal{LZ}$,
$\mathcal{D}=\mathcal{ZM}\vee\mathcal{SL}$,
$\mathcal{A}=\mathcal{ZM}\vee\mathcal{LRB}$,
$\mathcal{LNB}=\mathcal{LZ}\vee\mathcal{SL}$, we get

$\mathcal{C}\vee\mathcal{LRB}=\mathcal{B}\vee\mathcal{D}\vee\mathcal{LRB}=\mathcal{LZ}\vee\mathcal{ZM}\vee\mathcal{ZM}\vee\mathcal{SL}\vee\mathcal{LRB}=$
$=\mathcal{ZM}\vee\mathcal{LRB}\vee\mathcal{LZ}\vee\mathcal{SL}=\mathcal{A}\vee\mathcal{LNB}=\mathcal{A},$

because $\mathcal{LNB}\subseteq\mathcal{A}$.

$\mathcal{A}=\mathcal{B}\vee\mathcal{LRB}$: Using equations
$\mathcal{B}=\mathcal{ZM}\vee\mathcal{LZ}$ and
$\mathcal{A}=\mathcal{ZM}\vee\mathcal{LRB}$, we get

$\mathcal{B}\vee\mathcal{LRB}=\mathcal{LZ}\vee\mathcal{ZM}\vee\mathcal{LRB}=\mathcal{LZ}\vee\mathcal{A}=\mathcal{A},$

because $\mathcal{LZ}\subseteq\mathcal{A}$. ∎

An element $a$ of a lattice $(L;\vee,\wedge)$ with a zero $0$ is said to be an
_atom_ of $L$ if $0<a$ and $0\leq b<a$ implies $0=b$ for arbitrary $b\in L$.
An element $a$ of a lattice $(L;\vee,\wedge)$ is called a _neutral element_
[4] of $L$ if, for all $x,y\in L$, the triple $\\{a,x,y\\}$ generates a
distributive sublattice of $L$. It is known [10] that an element $a$ of a
lattice $L$ is a neutral element if and only if, for all $x,y\in L$, $(a\wedge
x)\vee(a\wedge y)\vee(x\wedge y)=(a\vee x)\wedge(a\vee y)\wedge(x\vee y)$.

######  Theorem 15.

The varieties $\mathcal{ZM}$, $\mathcal{LZ}$, $\mathcal{SL}$ are atoms, the
varieties $\mathcal{T}$, $\mathcal{ZM}$, $\mathcal{SL}$,
$\mathcal{D}=\mathcal{ZM}\vee\mathcal{SL}$, and $\mathcal{LLS}$ are neutral
elements of the lattice ${\mathcal{L}}({\bf LLS})$.

###### Proof.

By [11, Theorem 2.6], $\mathcal{ZM}$, $\mathcal{LZ}$ and $\mathcal{SL}$ are
atoms of the lattice of all semigroup varieties (see also the Theorem of
Section IV of [8]). Thus they are atoms of the lattice ${\mathcal{L}}({\bf
LLS})$. It is clear that the varieties $\mathcal{T}$ and $\mathcal{LLS}$ are
neutral elements of the lattice ${\mathcal{L}}({\bf LLS})$. By [27,
Proposition 4.1], varieties $\mathcal{ZM}$, $\mathcal{SL}$, and
$\mathcal{D}=\mathcal{ZM}\vee\mathcal{SL}$ are neutral elements of the lattice
of all semigroup varieties. Thus they are neutral elements of the lattice
${\mathcal{L}}({\bf LLS})$. ∎

The previous results on the lattice ${\mathcal{L}}({\bf LLS})$ is illustrated
by Fig 1.

$\mathcal{{LZ}}$$\bullet$$\mathcal{{T}}$$\bullet$$\mathcal{{ZM}}$$\bullet$$\mathcal{{SL}}$$\bullet$$\mathcal{B}$$\bullet$$\mathcal{D}$$\bullet$$\mathcal{C}$$\bullet$$\mathcal{LNB}$$\bullet$$\mathcal{LRB}$$\bullet$$\mathcal{LLS}$$\mathcal{A}$$\bullet$
Figure 1:

The varieties on the lower level of the diagram are the left regular band
varieties (see [8, Figure 4]). Their number is finite. However, the question
of how many elements are in the lattice of all left legal semigroup varieties
has not yet been answered. For example, the semigroup $S$ defined in Example 4
is a left legal semigroup, but it is not included in any of the proper
subvarieties of $\mathcal{LLS}$ on Fig. 1. The following problem needs a
solution.

OPEN PROBLEM: Is the lattice of all left legal semigroup varieties finite?

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