_{1}

^{*}

The internal Zappa-Szép products emerge when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. The external version constructed from actions of two semigroups on one another satisfying axiom derived by G. Zappa. We illustrate the correspondence between the two versions internal and the external of Zappa-Szép products of semigroups. We consider the structure of the internal Zappa-Szép product as an enlargement. We show how rectangular band can be described as the Zappa-Szép product of a left-zero semigroup and a right-zero semigroup. We find necessary and sufficient conditions for the Zappa-Szép product of regular semigroups to again be regular, and necessary conditions for the Zappa-Szép product of inverse semigroups to again be inverse. We generalize the Billhardt λ-semidirect product to the Zappa-Szép product of a semilattice E and a group G by constructing an inductive groupoid.

The Zappa-Szép product of semigroups has two versions internal and external. In the internal one we suppose that S is a semigroup with two subsemigroups A and B such that each

For groups, the two versions are equal, but as we show in this paper for semigroups this is true for only some special kinds of semigroups.

Zappa-Szép products of semigroups provide a rich class of examples of semigroups that include the self- similar group actions [^{*}-algebras to associate a C^{*}-algebra to Zappa-Szép products and gives an explicit presentation of the algebra. They define a quotient C^{*}-algebra that generalises the Cuntz-Pimsner algebras for self-similar actions. They specifically discuss the Baumslag-Solitar groups, the binary adding machine, the semigroup

In [

In [

In this paper we give general definitions of the Zappa-Szép product and include results about the Zappa-Szép product of groups and a special Zappa-Szép product for a nilpotent group.

We illustrate the correspondence between the internal and external versions of the Zappa-Szép product. In addition, we give several examples of both kinds. We consider the structure of the internal Zappa-Szép product as an enlargement. We show how a rectangular band can be described as the Zappa-Szép product of a left-zero semigroup and a right-zero semigroup.

We characterize Green’s relations (

We construct from the Zappa-Szép product of a semilattice E and a group G, an inverse semigroup by constructing an inductive groupoid.

We rely on basic notions from semigroup theory. Our references for this are [

Let S be a semigroup with subsemigroups A and B such that each element

Thus the product in S can be described in terms of the two functions. Using the associativity of the semigroup S and the uniqueness property, we deduce the following axioms for the two functions. By the associativity of S, we have

Now

and

Thus, by uniqueness property, we have the following two properties

Similarly by the associativity of S, we have

Now

and

Thus, by uniqueness property, we have the following two properties

In the following we illustrate which subsemigroups may be involved in the internal Zappa-Szép product.

Lemma 1. If the semigroup S is the internal Zappa-Szép product of A and B then

Proof. Consider

Of course, if S is a monoid and A and B are submonoids then

Proposition 1. If

Proof. We use Brin’s ideas in [

Therefore

Similarly

Therefore

Set

Hence

Hence

Lemma 2. Let

Proof. We have

Similarly, since

In an internal Zappa-Szép product

In the following we give a definition of the enlargement of a semigroup introduced in [

Definition 1. A semigroup T is an enlargement of a subsemigroup S if

Example 1. [

Proposition 2. Let S be the internal Zappa-Szép product of subsemigroups A and B. Then S is an enlargement of a local submonoid eSe for some

monoids

Proof. We have

We note that if

Following [

can find

Thus

defined by

In this Section we give some general properties of the Zappa-Szép product. We characterize Green’s relations (

Proposition 3. [

(i)

(ii)

Proof. Suppose

and

Hence

It follows that

Proposition 4. In the Zappa-Szép product

Proof. By Proposition 3 we have _{1} and z_{2} in M such that

Then

Therefore we set

Similarly

But from the following example we conclude that the action of the group G is a group action is a necessary condition.

Example 2. Let

Let

Observe that

Thus Zappa-Szép axioms are satisfied, since define

is a morphism (this is easy to see from the fact that

where

and

We therefore have

For

For

For

and

Thus

since

since

Now, we note that

But

But

and so

Proposition 5. Let

Proof. Suppose that

Then

which implies

and

which implies

Thus by (1) and (2) we have

Now suppose

Therefore we set

Proposition 6. If

in

Proof. Suppose

We set

Similarly

The main goal of this Section is to determine some of the algebraic properties of Zappa-Szép products of semigroups in terms of the algebraic properties of the semigroups themselves.

The (internal) Zappa-Szép product

Example 3. Let

Example 4. Take

However, there are criteria we can prove that the internal Zappa-Szép product

Proposition 7. If A is a regular monoid, B is a group,

Proof. Let

then

whereupon

Proposition 8. Let A be a left zero semigroup and B be a regular semigroup. Suppose that for all

Proof. Let

and

Theorem 1. [

We now give a general necessary and sufficient condition for Zappa-Szép products of regular semigroups to be regular. Consider the internal Zappa-Szép product

Theorem 2. Let A and B be regular subsemigroups and

Proof. Given

is not empty, see [

and

and so

and so

Thus

we have

Then

and

and so

Conversely, if

Corollary 1. If A and B are regular and

Proof. If we take

In this case: if

Now we discuss inverse Zappa-Szép products. Let S and T be inverse semigroups/monoids with

Example 5. [

A complete characterization of semidirect products of monoids which are inverse monoids is given in Nico [

Theorem 3. [

In the general case of the Zappa-Szép product of inverse semigroups

Example 6. Let

and

and

and

We check Zappa-Szép axioms by the following: define

This is a homomorphism of groups since

We have a homomorphism

We note that

Since

and

So

and

Thus

The achievement of necessary and sufficient conditions was difficult; so we try to find an inverse subset of the Zappa-Szép product of inverse semigroups. This achieved and described in Section 9. We have given the necessary conditions for Zappa-Szép products of inverse semigroups to be inverse in the following theorem.

Theorem 4.

(i) S and T are inverse semigroups;

(ii)

(iii) For each

Proof. We know that

Thus

and

By (iii) a and t act trivially on each other, b and u act trivially on each other, then

But since S and T are inverse semigroup, then idempotents commutes that is

Then

Thus

Let A and B be semigroups, and suppose that we are given functions

If A and B are semigroups that both have zero elements (

Then by Proposition [

Example 7. If A and B are semigroups with

From the following example we deduce that the zeros 0_{A} and 0_{B} of A and B respectively do not necessarily give a zero for the external Zappa-Szép product

Example 8. If A is a monoid with identity

The Zappa-Szép rules can be demonstrated using a geometric picture: draw elements from A as horizontal arrows and elements from B as vertical arrows. The rule

From the horizontal composition we get

From the vertical composition we get (ZS1) and (ZS4) as follows:

These pictures show that a Zappa-Szép product can be interpreted as a special kind of double category. This viewpoint on Zappa-Szép products underlies the work of Fiedorowicz and Loday [

In general, there is not a perfect correspondence between the internal and external Zappa-Szép product of semi- groups. For one thing, embedding of the factors might not be possible in an external product as the following example demonstrates.

Example 9. Consider the external Zappa-Szép product

a subgroup of P isomorphic to

However, under some extra hypotheses, the external product can be made to correspond to an internal product for example:

・ if we assume the two factors A and B involving in the external Zappa-Szép product have an identities ele- ments

So if

_{M} is a homo-

morphism, also

・ If A is a left zero semigroup and B is a right zero semigroup, then the external Zappa-Szép product of A and B is a rectangular band and it is the internal Zappa-Szép product of

Theorem 5. M is the internal Zappa-Szép product of a left-zero semigroup A and a right-zero semigroup B if and only if M is a rectangular band.

Proof. Let A be a left-zero semigroup and B a right-zero semigroup. In the rectangular band

Conversely, Let

Thus M is a rectangular band. W

1) Let

2) Suppose that A is a band. Then the left and right regular actions of A on itself allows us to form the Zappa- Szép product

3) Let

4) For groups,

on B and

In this section we consider a particular Zappa-Szép product for nilpotent groups. Note that G being nilpotent of class at most 2 is equivalent to the commutator subgroup

In the following we show that these actions let us form a Zappa-Szép product

Proposition 9. Let G be nilpotent group of class at most 2. Then the left and right conjugation actions of G on it self can be used to form the Zappa-Szép product

Proof. Let G act on itself by left and right conjugation as follows:

where

We prove that the Zappa-Szép rules are satisfied if G is a nilpotent group of class less than or equal 2, which implies that

since G is nilpotent of class £ 2,then

Thus

since G is nilpotent of class 2, then

Thus

Proposition 10. If the left and right conjugation actions of G on itself satisfy the Zappa-Szép rules, then G is nilpotent of class at most 2.

Proof. Suppose the Zappa-Szép rules satisfied, we prove that G is nilpotent of class £ 2. If

then for all

Therefore

Combining Propositions 9 and 10 we prove the following:

Proposition 11. P is the Zappa-Szép product of the group G and G with left and right conjugation actions of G on itself if and only if G is nilpotent of class at most 2.

Next we prove the following:

Lemma 3. The center of

Proof. Suppose

and

Put in (1)

Lemma 4. If

Proof. If G is abelian then G is nilpotent of class 1 if and only if

If P is abelian then

Proposition 12. If P is the non-abelian Zappa-Szép product

Proof. We have G is nilpotent group of class £ 2 if and only if for all

Now

and

Write

Since

This implies that

Combining Propositions 11, 12 and Lemma 4 we have the following.

Theorem 6. Let G be a group that is nilpotent of class at most 2, and let

1) P is abelian if and only if G is abelian, in which case

2) If G is non-abelian and hence nilpotent of class 2, then P is also nilpotent of class 2.

The Zappa-Szép product of inverse semigroups need not in general be an inverse semigroup. This is even the case for the semidirect product as we see (Nico [

Note that if

We consider the following where E a semilattice and G a group, and subset

We form a groupoid from the action of the group G on the set E which has the following features:

・ vertex set:

・ arrow set:

・ an arrow

・ the inverse of the arrow

・ the identity arrow at e is

・ an arrows

is

Lemma 5. If

Proof. we have

Lemma 6. Suppose that

Proof. We have

and

Thus

Proposition 13. If

with composition defined by

if

Proof.

We have to prove

&

Now

Since

Then

&

But

&

But

Thus

Now we introduce an ordering on

Lemma 7. The ordering on

is transitive.

Proof. We have to prove that if

Lemma 8. The ordering on

is antisymmetric.

Proof. We have to prove if

Proposition 14.

is a partial order set.

Proof. Clear from the definition of the ordering that £ is reflexive. By Lemma 7 and Lemma 8 £ is transitive and antisymmetric. Thus

Next we prove that

Lemma 9. If

Proof. Suppose that

Thus

and

Therefore

and hence

Lemma 10. If

for all

Proof. Suppose that

Then we have

and

and we have the following

where

where

and

Then

Lemma 11. If

restriction of

Proof. Suppose

Moreover,

Thus

Proposition 15.

Proof. We prove that

lattice

Theorem 7. If

is an inverse semigroup with multiplication defined by

Proof. Let

the pseudoproduct

and

and

Now, since

Therefore

Now, we have in the ordering defined on

implies that

and

Therefore

Thus

We summarize the main results of this paper in the following:

1) We characterize Green’s relations (

2) We prove that the internal Zappa-Szép product S of subsemigroups A and B is an enlargement of a local submonoid eSe for some

3) We give the necessary and sufficient conditions for the internal Zappa-Szép product

4) The Zappa-szep products

5) The Zappa-Szép product of inverse semigroups need not in general be an inverse semigroup. In this paper we give the necessary conditions for their existence and we modified the definition of semidirect products in the inverse case to obtain what we termed