# Algebraic properties of Zappa–Szép products of semigroups and monoids

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## Abstract

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for these Zappa–Szép products. We consider a left restriction semigroup *S* with semilattice of projections *E* and define left and right actions of *S* on *E* and *E* on *S*, respectively, to form the Zappa–Szép product \(E \bowtie S\). We further investigate properties of \(E \bowtie S\) and show that *S* is a retract of \(E\bowtie S\). We also find a subset *T* of \(E \bowtie S\) which is left restriction.

## Keywords

Left restriction semigroups Generalised Green’s relations Semidirect products Zappa–Szép products## 1 Introduction

Semidirect products and their generalisation Zappa–Szép products have become well known tools to decompose algebraic structures such as semigroups over recent years. The aim of this paper is to study algebraic properties of Zappa–Szép products of semigroups. The Zappa–Szép products of semigroups were properly developed by Kunze [14] who gave applications of Zappa–Szép products to translational hulls, Bruck–Reilly extensions and Rees matrix semigroups. The Zappa–Szép products of semigroups involve actions of two semigroups on each other. These actions are closely related to the actions of Mealy machines (automata with output). For convenience of the reader we begin by recalling the basic definition of Zappa–Szép products of semigroups.

### Definition 1.1

*S*and

*T*be semigroups and suppose that we have maps

*T*acts on

*S*from the left and

*S*acts on

*T*from the right, respectively. Define a binary operation on \(S\times T\) by

*S*and

*T*and denoted by \(S\bowtie T\).

Note that if one of the above actions is trivial (that is, one semigroup acts by the identity map), then the second action is by morphisms, and we obtain the semidirect product \(S\rtimes T\) (if *S* acts trivially) or \(S\ltimes T\) (if *T* acts trivially).

*S*and

*T*are monoids then the following four axioms are also required to hold:

*Z*is a monoid with submonoids

*S*and

*T*such that \(Z=ST\) and every element \(z\in Z\) has a unique decomposition as \(z=st\), where \(s\in S\) and \(t\in T\), then we say that

*Z*is the

*internal*Zappa–Szép product of

*S*and

*T*. If \(Z=S\bowtie T\) is the external Zappa–Szép product of monoids

*S*and

*T*, then putting

*Z*, isomorphic to

*S*and

*T*respectively, such that

*Z*is the internal Zappa–Szép product of \(S^{\prime }\) and \(T^{\prime }\). Conversely if \(Z=ST\) is the internal Zappa–Szép product of submonoids

*S*and

*T*then uniqueness of decompositions and associativity enable us to show that

*Z*is isomorphic to an external Zappa–Szép product \(S\bowtie T\). For semigroups there is no such universal correspondence between internal and external Zappa–Szép products and indeed not even for semidirect products, as remarked by Preston [20]. Preston went on to provide, by judicious adding of identities, a way to move between internal and external semidirect products of semigroups, but the correspondence is weaker than that in the case for monoids [20]. A similar correspondence in the case of Zappa–Szép products of semigroups is given in the thesis of the author [26].

In Sect. 2 we record the behaviour of the generalised Green’s relations on the Zappa–Szép products of semigroups and monoids. Green’s relations \(\mathcal R\), \(\mathcal L\), \(\mathcal H\), \(\mathcal {D}\) and \(\mathcal J\) are well known equivalence relations that characterise the elements of a semigroup in terms of one and two-sided principal ideals they generate. The reader is referred to [12] for further details concerning Green’s relations. We explain how generalised Green’s relations may be used to characterise elements of non-regular semigroups.

*S*is defined by the rule that for any \(s,t\in S\), \(s\, {\mathcal R}^{*}\, t\) (\(s\, {\mathcal L}^{*}\, t\)) if and only if for all \(u,v\in S^{1}\) (the extension of

*S*by the addition of an identity element)

*e*of

*S*acts as a left identity for its \({\mathcal R}^{*}\)-class and a right identity for its \({\mathcal L}^{*}\)-class. Also note that \(\mathcal R\subseteq {\mathcal R}^{*}\) and \(\mathcal L\subseteq {\mathcal L}^{*}\). If

*S*is regular then \(\mathcal R= {\mathcal R}^{*}\) and \(\mathcal L= {\mathcal L}^{*}\). A useful observation is that unlike Green’s relations, the relations \({\mathcal R}^{*}\) and \({\mathcal L}^{*}\) need not commute.

*left (right, two-sided) restriction*semigroup. Let

*S*be a semigroup and let

*E*be a subset of

*E*(

*S*), where

*E*(

*S*) denotes the set of idempotents of

*S*. The relation \(\widetilde{\mathcal {R}}_E\) on

*S*is defined as for any \(s,t\in S\), \(s\, \widetilde{\mathcal {R}}_E\, t\) if and only if for all \(e\in E\),

*s*and

*t*have same set of left identities in

*E*. Dually, \(s\, \widetilde{\mathcal {L}}_E\, t\) if and only

*s*and

*t*have same set of right identities in

*E*.

Certainly \(\widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) are equivalence relations, however, unlike the case for \(\mathcal R\), \(\mathcal L\), \({\mathcal R}^{*}\) and \({\mathcal L}^{*}\), it is not true that \(\widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) are respectively left and right compatible.

### Lemma 1.2

*S*be a semigroup and \(E\subseteq E(S)\). If \(s\in S\) and \(e\in E\), then \(s\, \widetilde{\mathcal {R}}_E\, e\) (\(s\, \widetilde{\mathcal {L}}_E\, e\)) if and only if \(es=s\) (\(se=s\)) and for all \(f\in E\),

It is easy to see that \(\mathcal R\subseteq {\mathcal R}^{*}\subseteq \widetilde{\mathcal {R}}_E\) and \(\mathcal L\subseteq {\mathcal L}^{*}\subseteq \widetilde{\mathcal {L}}_E\). If *S* is regular and \(E=E(S)\), then the foregoing inclusions are replaced by equalities. More generally, if \(e,f\in E\) then \(e\,\widetilde{\mathcal {R}}_E\, f\) if and only if \(e\,\mathcal R\, f\) and \(e\,\widetilde{\mathcal {L}}_E\, f\) if and only if \(e\,\mathcal {L}\, f\). In general, however, the inclusions are strict.

*s*by \(s^+\) and \(s^*\), respectively. Note that \(s^+\) and \(s^*\) are unique if

*E*is a semilattice. We note that for any distinguished idempotents \(s^+\) and \(s^*\) in \(\widetilde{\mathcal {R}}_E\)-class and \(\widetilde{\mathcal {L}}_E\)-class of

*s*respectively, we have that

*left restriction semigroup*and a semilattice. Restriction semigroups and their one-sided versions have been developed from various points of view and under different names since the 1960s. Left restriction semigroups appeared in their own right for the first time in the work of Trokhimenko [24]. They were studied as

*type SL2*\(\gamma \)-

*semigroups*in early 1980s by Batbedat and Fountain [1, 2]. Lawson studied (left) restriction semigroups as the idempotent connected Ehresmann semigroups by drawing connection between semigroup theory and category theory [15]. In the last decade they were studied by Jackson and Stokes [13] in the guise of

*(left) twisted*

*C*-

*semigroups*motivated by consideration of closure operators, and by Manes [16] as

*guarded semigroups*which arose from the restriction categories of Cockett and Lack [4].

In recent years, an analogous theory to that of structure theory of inverse semigroups has been developed in the broader setting of restriction semigroups. One of the fundamental results in the structure theory of semigroups is that every inverse semigroup is an idempotent separating homomorphic image of a ‘proper’ inverse semigroup where a proper inverse semigroup is isomorphic to a so called ‘*P*-semigroup’ and embeds into a semidirect product of a semilattice by a group [17, 18, 19]. For (left) restriction semigroups, many authors have produced work using constructions similar (at least on the surface) to those of McAlister, replacing groups by monoids of various kinds. Covering results for restriction semigroups are known due to Cornock, Fountain, Gomes and Gould [3, 5, 7, 8]. The structure of proper restriction semigroups is determined by analogues of *P*-semigroups. It has been shown that every restriction semigroup is embeddable into an almost left factorisable restriction semigroup. Also every restriction semigroup has a cover which is embeddable into a *W*-product (a special kind of semidirect product) [22, 23]. Recent development in this theory reveals that restriction semigroups have become a topic of lively research interest and in providing the common frame work for answering different questions in algebra, they will continue to be a key topic.

*left restriction semigroup*

*S*is a unary semigroup \((S,\cdot ,^+)\), where \((S,\cdot )\) is a semigroup and \(^+\) is a unary operation such that the following identities hold:

*E*is a semilattice (semigroup with commuting idempotents). We call

*E*the

*semilattice of projections (formerly distinguished semilattice)*of

*S*.

*right restriction semigroups*are defined where in this case the unary operation is denoted by \(^*\). A

*restriction semigroup*is a bi-unary semigroup

*S*which is both left restriction and right restriction and which also satisfies the linking identities

A left restriction semigroup *S* is equipped with a partial order \(\le \) defined by the rule that for any \(s,t\in S\), \(s\le t \) if and only if \( s=s^+t\) [11]. It is easy to check that \(\le \) is compatible on both sides in *S*. The following technical result will be useful in the subsequent sections of this paper.

### Lemma 1.3

*S*is left restriction semigroup with semilattice of projections

*E*, then for any \(s,t\in S\),

- (i)
\((s^+)^+=s^+\);

- (ii)
\((st)^+\le s^+\);

- (iii)
\((st^+)^+=(st)^+\).

We refer the reader to [11] for a detailed study on left (right, two-sided) restriction semigroups.

The following lemma provides an alternative characterisation for left restriction semigroups.

### Lemma 1.4

*S*is left restriction with semilattice of projections

*E*if and only if

- (i)
*E*is a semilattice; - (ii)
every \(\widetilde{\mathcal {R}}_E\)-class contains an idempotent of

*E*; - (iii)
the relation \(\widetilde{\mathcal {R}}_E\) is a left congruence and

- (iv)
the left ample condition holds, that is, for all \(s\in S\) and \(e\in E\),

The dual result holds for right restriction semigroups and the relation \(\widetilde{\mathcal {L}}_E\).

## 2 Generalised Green’s relations on Zappa–Szép products

Green’s relations \(\mathcal R\) and \(\mathcal {L}\) on a Zappa–Szép product of semigroups are characterised by Kunze [14]. We aim to characterise generalised Green’s relations in this section. We consider Zappa–Szép products of semigroups and monoids and record the behaviour of the relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\). The following is immediate from the definitions.

### Lemma 2.1

*S*and

*T*, then \((a,b)\, {\mathcal R}^{*}\, (c,d)\) if and only if for all \(x,\, u\in S\) and for all \(y,\, v\in T\)

### Proposition 2.2

*S*and

*T*. Then the following hold:

- (1)
if \(b,\, d\in T\) and \(b\, {\mathcal R}^{*}\, d\), then \((a,b)\, {\mathcal R}^{*}\, (a,d)\) in

*Z*for \(a\in S\); - (2)
if \(a,\, c\in S\) and \(a\, {\mathcal L}^{*}\, c\), then \((a,b)\, {\mathcal L}^{*}\, (c,b)\) in

*Z*for \(b\in T\).

### Proof

*Z*.

The proof of (2) is dual.\(\square \)

### Proposition 2.3

*S*,

*T*be monoids and \(Z=S\bowtie T\) be the Zappa–Szép product of

*S*and

*T*. Then

- (1)
\((a,b)\, {\mathcal R}^{*}\,(c,d)\) in

*Z*implies \(a\, {\mathcal R}^{*}\, c\) in*S*; - (2)
\((a,b)\, {\mathcal L}^{*}\, (c,d)\) in

*Z*implies \(b\, {\mathcal L}^{*}\, d\) in*T*.

### Proof

*Z*. To show that \(a\, {\mathcal R}^{*}\, c\) in

*S*, let \(x,u\in S\) be such that

*S*.

The proof for \({\mathcal L}^{*}\) is dual.\(\square \)

### Definition 2.4

*S*and

*T*be semigroups and \(\alpha : S\rightarrow T\) be a map from

*S*to

*T*. We define \({\text {Ker}}\alpha \) to be the relation

For more complete results, we now consider the reverse semidirect product \(S\ltimes T\) of *S* and *T* and use Lemma 2.1 in the context when one of the actions is trivial, because in the Zappa–Szép product \(S\bowtie T\) of monoids (semigroups) *S* and *T* if one of the actions is trivial (that is, one semigroup acts by the identity map), then the second action is by morphisms, and we obtain the semidirect product \(S\rtimes T\) (if *S* acts trivially) or \(S\ltimes T\) (if *T* acts trivially).

### Proposition 2.5

Suppose \(Z=S\ltimes T\) is a semidirect product of monoids *S* and *T*, where *T* is right cancellative. Also suppose that for any \(a,\, c\in S\), \(a\, {\mathcal R}^{*}\, c\) implies \({\text {Ker}}a\, =\, {\text {Ker}}c\) (where \({\text {Ker}}a\) is the kernel of the map induced by the right action of a). Then \(a\, {\mathcal R}^{*}\, c\) in *S* implies that \((a,b)\, {\mathcal R}^{*}\, (c,d)\) in *Z* for all \(b,d\in T\).

### Proof

*S*and let \(x,u\in S\) and \(y,v\in T\). Then

*Z*.\(\square \)

### Corollary 2.6

Suppose \(Z=S\ltimes T\) is a semidirect product of monoids *S* and *T*, where *T* is right cancellative. Also suppose that *S* acts on *T* injectively. Then \(a\, {\mathcal R}^{*}\, c\) in *S* implies that \((a,b)\, {\mathcal R}^{*}\, (c,d)\) in *Z* for all \(b,d\in T\).

### Definition 2.7

Let *M* be a monoid and *X* be a set. An action of *M* on *X* is called *faithful* if for \(m,n\in M\) with \(m\ne n\), there exists an \(x\in X\) such that \(m\cdot x\ne n\cdot x\).

### Theorem 2.8

Let \(Z=S\bowtie T\) be a Zappa–Szép product of monoids *S* and *T*, where T is right cancellative. Suppose *S* acts faithfully on the right of *T* and for any \(a,\, c\in S\), \(a\, {\mathcal R}^{*}\, c\) implies \({\text {Ker}}a\, =\, {\text {Ker}}c\). Then \(a\, {\mathcal R}^{*}\, c\) in *S* implies that \((a,b)\, {\mathcal R}^{*}\, (c,d)\) in *Z* for all \(b,d\in T\).

### Proof

*S*. To show that \((a,b)\, {\mathcal R}^{*}\, (c,d)\) in

*Z*, let \(x,u\in S\) and \(y,v\in T\) be such that

*T*is right cancellative. Also as \({\text {Ker}}a={\text {Ker}}c\), we have \(y^c=v^c\) and thus

*T*is right cancellative, therefore

*S*acts faithfully, we have

*Z*.\(\square \)

### Theorem 2.9

*S*,

*T*be monoids and let \(Z=S\bowtie T\) be a Zappa–Szép product of

*S*and

*T*. Let \(E\subseteq E(S)\) and \(F\subseteq E(T)\). Put

*Z*and

- (1)
\((a,b)\, \widetilde{\mathcal {R}}_{\overline{E}}\, (c,d)\) in

*Z*if and only if \(a\, \widetilde{\mathcal {R}}_{E}\, c\) in*S*; - (2)
\((a,b)\, \widetilde{\mathcal {L}}_{\overline{F}}\, (c,d)\) in

*Z*if and only if \(b\, \widetilde{\mathcal {L}}_{F}\, d\) in*T*.

### Proof

It is easy to check that \(\overline{E}\cup \overline{F}\subseteq E(Z)\).

*Z*. Let \(e\in E\). Then

*S*.

*S*and let \((a,b)\in Z\). Then for \((e,1)\in \overline{E}\) we have

The proof for \(\widetilde{\mathcal {L}}_{E}\) is dual.\(\square \)

### Theorem 2.10

*S*and

*T*be semigroups and \(Z=S\bowtie T\) be a Zappa–Szép product of

*S*and

*T*. Let \(E\subseteq E(S)\) and \(F\subseteq E(T)\). Suppose there exists a right identity

*e*for

*S*and a left identity

*f*for

*T*such thatPut

*Z*and

- (i)
\((a,b)\, \widetilde{\mathcal {L}}_{\overline{F}} \, (c,d) \Rightarrow b\, \widetilde{\mathcal {L}}_{F} \, d\) and \((a,b)\, \widetilde{\mathcal {R}}_{\overline{E}} (c,d)\, \Rightarrow a\, \widetilde{\mathcal {R}}_{E} \, c\);

- (ii)
\((a,b)\, {\mathcal L}^{*}\, (c,d)\Rightarrow b\, {\mathcal L}^{*}\, d\) and \((a,b)\, {\mathcal R}^{*}\, (c,d)\Rightarrow a\, {\mathcal R}^{*}\, c\).

### Proof

*Z*. For this let \((h,f)\in \overline{E}\). Then

Dually \((a,b)\, \widetilde{\mathcal {R}}_{\overline{E}} \, (c,d)\) implies \( a\, \widetilde{\mathcal {R}}_{E} \, c\).

Dually \((a,b)\, {\mathcal L}^{*}\, (c,d)\) implies that \(b\, {\mathcal L}^{*}\, d\).\(\square \)

## 3 Zappa–Szép product of a left restriction semigroup and a semilattice

In this section we consider a left restriction semigroup with semilattice of projections *E*. By defining a left action of *S* on *E* and a right action of *E* on *S*, we see that \(E\bowtie S\) becomes a Zappa–Szép product. This result was mentioned without proof in [10], as an example of a Zappa–Szép product. We include the full argument in this paper for the sake of completeness. Also we notice that \(E\bowtie S\) is not itself left restriction but it contains a subsemigroup *T* which is left restriction. Moreover, *S* is a retract of \(Z=E\bowtie S\).

### Lemma 3.1

Let *S* be a left restriction semigroup with semilattice of projections *E*. Define an action of *S* on *E* by \(s\cdot e=(se)^+\) and an action of *E* on *S* by \(s^{e}=se\). Then \(Z=E\bowtie S\) is a Zappa–Szép product of *S* and *E*.

### Proof

We check that these two actions satisfy the axioms of a Zappa–Szép product.

*E*acts on

*S*trivially, that is, \(s^e=s\), then

*S*acts on

*E*by morphisms as

If *S* is left restriction, then the Zappa–Szép product \(E\bowtie S\) obtained as above is the *standard Zappa–Szép product* of *E* and *S*.

We now compute the set of idempotents of \(Z=E\bowtie S\) .

### Lemma 3.2

*S*be a left restriction semigroup with semilattice of projections

*E*. Suppose that \(Z=E\bowtie S\) is the standard Zappa–Szép product of

*S*and

*E*. Then

*E*and if \(E(S)=E\), then \(\overline{E}=E(Z)\).

### Proof

*E*.

\(\square \)

We now record some of the properties of \(Z=E\bowtie S\).

### Proposition 3.3

*S*be a left restriction semigroup with semilattice of projections

*E*. Suppose that \(Z=E\bowtie S\) is the standard Zappa–Szép product of

*S*and

*E*. Let \((e,s)\in Z\), then:

- (a)
\(\overline{E} =\{(e,e):e\in E\}\) is a semilattice isomorphic to

*E*(*S*); - (b)
there is a semigroup epimorphism \(\alpha :Z\rightarrow S\) separating the idempotents of \(\overline{E}\);

- (c)
\((g,g)(e,s)=(e,s)\) if and only if \(ge=e\) and \(es=s\);

- (d)
(

*e*,*s*) has a left identity in \(\overline{E}\) if and only if \(es=s\); in this case \((e,s)\, \widetilde{\mathcal {R}}_{\overline{E} }\, (e,e)\). - (e)
\((e,s)(f,f)=(e,s)\) if and only if \( e\le s^+, s=sf\);

- (f)
for \((e,s)\in Z\), \((e,s)\, \widetilde{\mathcal {L}}_{\overline{E} }\, (f,f)\) where \((f,f)\in \overline{E}\) if and only if \( e\le s^+\) and \(s\, \widetilde{\mathcal {L}}_{E }\, f\);

- (g)
\((g,g)\, \widetilde{\mathcal {R}}_{\overline{E} }\, (e,s)\, \widetilde{\mathcal {L}}_{\overline{E} }\, (f,f)\) for some \(g,\, f\in E\) implies \((e,s)=(s^+,s)\).

*S*into \(Z=E\bowtie S\) under \(s\mapsto (s^+,s)\).

### Proof

(a) From Lemma 3.2, we know that \(\overline{E}\) is a semilattice isomorphic to *E*(*S*).

*e*,

*e*) is a left identity of (

*e*,

*s*) as

(g) It is straightforward to prove that \((g,g)\, \widetilde{\mathcal {R}}_{\overline{E} }\, (e,s)\, \widetilde{\mathcal {L}}_{\overline{E} }\, (f,f)\) for some \(g,f\in E\) implies that \((e,s)=(s^+,s)\) using (c) and (e).

*U*is a subsemigroup of

*Z*, let \((s^{+},s),(t^{+},t)\in U\), then

*U*is a left restriction subsemigroup of

*Z*where

We remark that our construction immediately applied to the regular case if *S* is an inverse semigroup as shown in [6].

The Zappa–Szép product \(Z=E\bowtie S\) of a left restriction semigroup *S* and *E* is not left restriction in general. The same is true for the semidirect product of a left restriction semigroup and a semilattice. But in the special case where *S* is a monoid (because monoids are left restriction), it is proved by Fountain et al. [5] that the semidirect product of a monoid with a semilattice is again left restriction. A similar result was proved by Manes [16] in the guise of *guarded semigroups* and by Gomes and Gould [7] in the case of unipotent monoids for *weakly left ample semigroups*, where a *weakly left ample semigroup* is a left restriction semigroup with \(E=E(S)\). All of these authors obtained covering theorems of McAlister type for their construction, namely there is an idempotent-separating homomorphism from a subsemigroup of the semidirect product of a monoid with a semilattice to a left restriction semigroup. Although the Zappa–Szép product of a semilattice by a left restriction semigroup is not left restriction in general, we nevertheless show that if *S* is left restriction with semilattice of projections *E*, then we may construct a Zappa–Szép product \(Z=E\bowtie S\) in a canonical way as in Proposition 3.3 such that there is an idempotent separating semigroup morphism from *Z* onto *S*. Indeed, *S* is a retract of \(E\bowtie S\), as is immediate from Proposition 3.3 [(b) and the final statement].

We have shown in Proposition 3.3 that *Z* contains the left restriction subsemigroup *U* with semilattice of projections \(\overline{E}\), which is isomorphic to *S*. We now find a left restriction subsemigroup *T* of *Z* containing *U*, which, in view of Proposition 3.3 (d), is clearly maximum with respect to semilattice of projections \(\overline{E}\).

### Theorem 3.4

*S*be a left restriction semigroup with semilattice of projections

*E*. Suppose that \(Z=E\bowtie S\) is the standard Zappa–Szép product of

*S*and

*E*. Let

*T*is the maximum left restriction subsemigroup of

*Z*with semilattice of projections \(\overline{E}\).

Let \(\beta \) be the restriction to *T* of the semigroup morphism \(\alpha : Z\rightarrow S\) in Proposition 3.3 . Then \(\beta :T\rightarrow S\) is a projection separating epimorphism of left restriction semigroups.

### Proof

*T*is a subsemigroup of \(Z=E\bowtie S\). For this let \((e,s),(f,t)\in T\). Then \(es=s, ft=t\), so that

*T*is subsemigroup of

*Z*. Clearly \(\overline{E} \subseteq T\).

To show that *T* is left restriction, let \((e,s)\in T\). Then \((e,s)^+\, \widetilde{\mathcal {R}}_{\overline{E} }\, (e,e)\) follows from Proposition 3.3 (c) and the fact that \(\overline{E}\subseteq T\).

*T*is left restriction with semilattice of projections \(\overline{E}\). The fact that

*T*is maximum with respect to these properties comes immediately from Proposition 3.3(d).

Now define \(\beta : T\rightarrow S\) by \((e,s)\beta =es\). Clearly \(\beta \) is an epimorphism separating the elements of \(\overline{E}\).\(\square \)

*T*to be

*proper*in general as

*T*contains an isomorphic copy of

*S*. With a different approach, retaining the same action of

*S*on

*E*, but

*E*acting trivially on

*S*, we obtain a semidirect product \(E\rtimes S\) and this approach yields a

*proper cover*of any left restriction semigroup as proved by Manes [16] (see also [5, 7]). Manes’ left restriction subsemigroup of \(E\rtimes S\) is given as follows:

*proper cover*for

*S*under \((e,s)\mapsto es\).

If \(Z=E\bowtie S\) is the Zappa–Szép product of an inverse semigroup *S* and its semilattice *E*, then \(T=\{(e,s):es=s\}\) is not inverse. However, from Proposition 3.3 there is a subsemigroup of *T* which is inverse.

### Corollary 3.5

Let \(Z=E(S)\bowtie S\) be the standard Zappa–Szép product of an inverse semigroup *S* and semilattice of idempotents *E*(*S*) in the sense of [6] and let \(T=\{(e,s):es=s\}\) be a subsemigroup of *Z*. Then there is a canonical embedding of *S* into *T* under \(s\mapsto (ss^{-1},s)\).

We end this paper with couple of examples of Zappa–Szép products.

The following example is inspired by a result that appeared in [6], where Gilbert and Wazzan considered the Zappa–Szép product \(E\bowtie G\) of a group *G* and a semilattice *E*. By using the natural involution in a group they were able to define a Zappa–Szép product \(G\bowtie E\). Left restriction semigroups and monoids do not possess a natural involution in the way that groups and indeed inverse semigroups do. To overcome this, we use the notion of a double action (introduced in [5]) and compatibility conditions (introduced in [3]).

### Example 3.6

*E*and a monoid

*T*. Define a new action of

*T*on

*E*as follows:

*T*on

*E*satisfies

*E*on

*T*by the following.

Our final example appeared in [10] in context of inverse semigroups.

### Example 3.7

Let *X* be a set and \(S=T=\mathcal {P}(X)\) (power set under union). Then *S* and *T* are left restriction semigroups where \(s^+=s\) and \(t^+=t\), for all \(s\in S\) and \(t\in T\). Also we note that \(S=T\) is a semilattice. For convenience we regard *S* as a semilattice and *T* as left restriction semigroup. Define action of *T* on the left of *S* by the rule \(t\cdot s=t\cup s\) and action of *S* on the right of *T* by \(t^s=t{\setminus } s\). One can easily verify that (ZS1)–(ZS4) hold and thus \(Z=S\bowtie T\) is a Zappa–Szép product.

## Notes

### Acknowledgements

The author is grateful to the Schlumberger Foundation for funding her Ph.D. studies, of which this paper forms a part. The author is also grateful to Professor Victoria Gould for her useful suggestions on this manuscript. Also the author would like to thank the referee of this paper whose useful comments helped her in improving the presentation of this article.

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