1. Introduction
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
can be written uniquely as
with
and
Then since
we have
with
and
determined uniquely by a and b. We write
and
Associativity in S implies that the functions
and
satisfy axioms first formulated by Zappa [1] . In the external version we assume that we have semigroups A and B and assume that we have maps
defined by
and a map
defined by
which satisfy Zappa axioms [1] .
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 [2] . Recently, [3] uses Li’s construction of semigroup C*-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
, and the
-semigroup
.
In [4] they study semigroups possessing E-regular elements, where an element a of a semigroup S is E-regular if a has an inverse
such that
lie in
. They also obtain results concerning the extension of (one-sided) congruences, which they apply to (one-sided) congruences on maximal subgroups of regular semigroups. They show that a reasonably wide class of
-simple monoids can be decomposed as Zappa-Szép products.
In [5] we look at Zappa-Szép products derived from group actions on classes of semigroups. A semidirect product of semigroups is an example of a Zappa-Szép product in which one of the actions is taken to be trivial, and semidirect products of semilattices and groups play an important role in the structure theory of inverse semigroups. Therefore Zappa-Szép products of semilattices and groups should be of particular interest. We show that they are always orthodox and
-unipotent, but are inverse if and only if the semilattic acts trivially on the group, that is when we have the semidirect product. In [5] we relate the construction (via automata theory) to the
-semidirect product of inverse semigroups devised by Billhardt.
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 (
and
) of the Zappa-Szép product
of a monoid M and a group G. We prove some results about regular and inverse Zappa-Szép product of semigroups.
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 [6] and [7] .
2. Internal Zappa-Szép Products
Let S be a semigroup with subsemigroups A and B such that each element
is uniquely expressible in the form
with
and
We say that S is the “internal” Zappa-Szép product of A and B, and write
Since
with
and
, we must have unique elements
and
so that
This defines two functions
and
Since
with
and
, we must have unique elements
and
so that
Write
and
This defines two function
and
Thus
Using these definitions, we have for all
and
that
![]()
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
(ZS2) ![]()
(ZS3) ![]()
Similarly by the associativity of S, we have
![]()
Now
![]()
and
![]()
Thus, by uniqueness property, we have the following two properties
(ZS1) ![]()
(ZS4) ![]()
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
Then since
we have
for all
for all
Thus x is a right identity for A and left identity for B, whereupon
. Observe that
thus
W
Of course, if S is a monoid and A and B are submonoids then ![]()
Proposition 1. If
the internal Zappa-Szép product of A and B, then ![]()
Proof. We use Brin’s ideas in [8] Lemma 3.4. If
then
for unique
and
giving us a function
and likewise, if
then
for some unique
and some function
But for all
we have
and therefore
and
: that is
is a left identity for B. Similarly,
and
is a right identity for A. In particular,
and
are idempotents. Now
![]()
Therefore
(1)
Similarly
![]()
Therefore
(2)
Set
and for any
in (1):
![]()
Hence
is constant:
for all
Similarly, setting
and
in (2):
![]()
Hence
is constant:
for all
But now we have that for all
and
and
But then putting
and
we have
and in particular
W
Lemma 2. Let
the internal Zappa-Szép product of A and B and
be a right identity for A and a left identity for B. Then
and ![]()
Proof. We have
then
, but
thus
and by uniqueness we have
![]()
Similarly, since
we have
. Also
Thus
Hence
and
W
In an internal Zappa-Szép product
we find an idempotent
This shows (for example) that a free semigroup cannot be a Zappa-Szép product. But in a monoid Zappa-Szép product
of submonoids A and B the special idempotent
must be
, since we have
uniquely. Then for all
and thus
and
Similarly
and ![]()
In the following we give a definition of the enlargement of a semigroup introduced in [8] for regular semigroups, and in [9] this concept is generalized to non-regular semigroups by describing a condition (enlarge- ment) under which a semigroup T is covered by a Rees matrix semigroup over a subsemigroup. We describe the enlargement concept for internal Zappa-Szép products.
Definition 1. A semigroup T is an enlargement of a subsemigroup S if
and
.
Example 1. [9] Let S be any semigroup and let I be a set of idempotents in S such that
. Then S is an enlargement of ISI because
and
If
and
then S is an enlargement of the local submonoid
.
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
and eSe is the internal Zappa-Szép product of the sub-
monoids
and
where ![]()
Proof. We have
such that e is a right identity for A and a left identity for B. Then
so
for
So S is an enlargement of the local submonoid
(
is a monoid with identity e). It is clear that
and
are submonoids of
. We must show that each element
is uniquely expressible as
with
If
then
But
for unique
and so
where
Since
and
this expression is unique, because
There- fore each element
is uniquely expressible as
with
W
We note that if
such that T is an enlargement of
, where
and
if
are regular with the assumption that if
then
and if
then
Then A, B are regular, since if
is regular monoid, then for each
there exists
such that
Now
which implies
Since A has a right identity e, then
. Similarly we get
. Thus A is regular. Similarly, we get B is regular.
Following [9] we describe the Rees Matrix cover for the Zappa-Szép product
such that T = TeT and is an enlargement of
for some idempotent
where
such that Ae = A and eB = B and S is the Zappa-Szép product of
and
with
and
Then by Corollary 4 in [9] the Rees matrix semigroup is given by
such that
since
For each
, we
can find
and
such that
So if
then
and
such that
Similarly, for
therefore
and
Now for each
fix elements
and define
matrix P by putting ![]()
Thus
is the Rees matrix cover for
where the map ![]()
defined by
is the covering map (
is a strict local isomorphism from M to T along which idempotents can be lifted).
3. Green’s Relations L and R on Zappa-Szép Products
In this Section we give some general properties of the Zappa-Szép product. We characterize Green’s relations (
and
) of the Zappa-Szép product
of a monoid M and a group G.
Proposition 3. [10] Let
be a Zappa-Szép product of semigroups A and B. Then
(i)
in B;
(ii)
in A.
Proof. Suppose
in
then there exist
such that
and. Then
![]()
and
![]()
Hence
![]()
It follows that
in B. Similar proof for (ii). W
Proposition 4. In the Zappa-Szép product
of a monoid M and a group G. Then
![]()
Proof. By Proposition 3 we have
implies that
in M. To prove the converse suppose that
in M then there exist z1 and z2 in M such that
and
To show that
we have to find
and
in
such that
![]()
![]()
Then
and
Hence
![]()
Therefore we set
Hence
![]()
Similarly
Hence
W
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
be a Clifford semigroup with the following multiplication table. Note that
and ![]()
Let
, the group of integers. Suppose that the action of
on A for each
is as follows:
![]()
Observe that
for all
. The action of A on
as follows:
![]()
Thus Zappa-Szép axioms are satisfied, since define
by
![]()
is a morphism (this is easy to see from the fact that
). Now define
(where
is the group of permutations on
) by
![]()
where
for all
. Clearly
is a morphism (of groups). Now for
and
we de- fine the action arises from the composition
as follows
![]()
and
![]()
We therefore have
and
as following.
For
and for ![]()
![]()
For
and using
for all
we have
![]()
For ![]()
![]()
and
![]()
Thus
the Zappa-Szép product of A and B. The set ![]()
since
and
if and only if
and ![]()
since
or
for all
so
or ![]()
Now, we note that
acts non-trivially. We have
and
in A but
not
-related to
where
since if we suppose
then there exist
and
in
such that
![]()
![]()
But
so
not
-related to
To calculate the
-class of
if
then
and so
or
we prove
is impossible so
If
for some
then we have
![]()
![]()
But
so
-related only to itself. Similarly
-related only to itself. To calculate the
-class of
suppose
then
or
Let
so
then
![]()
![]()
and so
which implies
or
so
Thus
for all
By similar calculation we have
So if
acts non-trivially we have a different structure for the
-classes of
and A.
![]()
Proposition 5. Let
be the semidirect product of a monoid M and a group G. Then
![]()
Proof. Suppose that
then there exist
in
such that
![]()
![]()
Then
and
Hence
there- fore
![]()
which implies
(1) ![]()
and
![]()
which implies
(2) ![]()
Thus by (1) and (2) we have
in M.
Now suppose
in G then there exist
and
in G such that
and
. Therefore
Hence
![]()
Therefore we set
in
other formula by symmetry
so
W
Proposition 6. If
such that
and
in G, then ![]()
in ![]()
Proof. Suppose
then there exist
and
in M such that
which implies that
![]()
We set
and
then
![]()
Similarly
Hence
in
W
4. Regular and Inverse Zappa-Szép Products
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
of the regular subsemigroups A and B need not to be regular in general. A special case of the Zappa-Szép product is the semidirect product for which one of the actions is trivial. We use Theorem 2.1 [11] to construct an example of regular submonoids such that their semidirect product is not regular.
Example 3. Let
be a commutative monoid with 0, each of whose elements is idempotent and such that
Let
be a monoid with two left zeros a and b. Then both S and T are regular semigroups. Let 1 acts trivially,
There is no
such that
for all
Thus the semidirect product
is not regular. For example,
is not a regular element of R.
Example 4. Take
such that
and
such that
Let
act trivially on B,
and
act trivially on A,
Then A and B are regular monoids but their Zappa-Szép product
is not regular.
However, there are criteria we can prove that the internal Zappa-Szép product
of regular A and B is regular as the following Propositions illustrated.
Proposition 7. If A is a regular monoid, B is a group,
for all
then
is regular. W
Proof. Let
where
and
we have to find
such that
. Now
and so we choose
where
Since we must have
but B is a group, so
Suppose we are given c, and choose
since B is a group, so
and
then
Since A is regular, we choose any
and set
thus Thus
whereupon
is regular. W
Proposition 8. Let A be a left zero semigroup and B be a regular semigroup. Suppose that for all
, there exists some
such that
and for all
there exists some
such that
Then
is regular.
Proof. Let
where
and
We have to find
such that
. Now
and we choose
where
Now since A is a left zero semigroup
and by our assumptions we can choose
such that ![]()
and
that is fixed by a. Then
Thus
is regular. W
Theorem 1. [12] For any arbitrary semigroup S,
is a subsemigroup of S if and only if the product of any pair of idempotents in S is regular.
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
of regular semigroups A and B. Then each
is uniquely a product of regular elements:
where
Hence M is regular if and only if
is a subsemigroup of M and so by Hall’s Theorem 1, M is regular if and only if the product of any two idempotents is regular. But in fact we need only consider products of idempotents
and
, as our next theorem shows.
Theorem 2. Let A and B be regular subsemigroups and
and
Then
if and only if
is regular.
Proof. Given
with
(uniquely) where
since A and B are regular subsemi- groups, then there exist
and
Then
and
Set
and
Then by the assumption
and the set
![]()
is not empty, see [14] . Because
let
, and let
Then
![]()
and
![]()
and so
Also
![]()
and so
Also
![]()
![]()
Thus
Also
because
![]()
we have
![]()
Then
![]()
and
![]()
and so
. Then
is a regular element which implies that
is regular.
Conversely, if
is the regular internal Zappa-Szép product of the regular subsemigroups A and B, each element m of M is uniquely written in the form
where
and
Thus if
and
this implies that
then
W
Corollary 1. If A and B are regular and
act trivially, then
is regular.
Proof. If we take
and
, then
is an idempotent in M. Because
since
act trivially. Therefore
Hence
is regular. W
In this case: if
, we can find
First find
and
![]()
since idempotents of A commute with those of B. Then
for some
Thus
where
and
Then
Now we discuss inverse Zappa-Szép products. Let S and T be inverse semigroups/monoids with
and let
be the semidirect product of S and T. We can see from the following example that P need not be inverse.
Example 5. [11] Let
be the commutative monoid with one non zero-identity idempotent a. Let
be the commutative monoid with zero an with
Then S and T are both inverse monoids, and there is a homomorphism
given by
and
Then
is regular. However, the element
has two inverses, namely
and
and hence P cannot be an inverse monoid.
A complete characterization of semidirect products of monoids which are inverse monoids is given in Nico [11] .
Theorem 3. [11] A semidirect product
of two inverse semigroups S and T will be inverse if and only if
acts trivially.
In the general case of the Zappa-Szép product of inverse semigroups
we have also P need not be inverse semigroup as we can see from the following example.
Example 6. Let
where
and
―Klein 4-group where
Suppose that the action of G on E is defined by:
![]()
and
![]()
and
acts trivially (that is each of
permutes
non-trivially). The action of E on G is defined by:
![]()
and
![]()
We check Zappa-Szép axioms by the following: define
by
![]()
This is a homomorphism of groups since
is the automorphism group of E. We have an action of G on E using
: if
and
define
![]()
We have a homomorphism
given by
and
The action of E on G is given by:
![]()
We note that
for all
For
we have for ![]()
![]()
Since
it is clear that
holds. For
we have for ![]()
![]()
and
![]()
So
holds and for ![]()
![]()
and
![]()
Thus
holds. Then
. Since every element of M is regular, then M is regular.
is a closed subsemigroup of M, so
is orthodox, but since
is not commutative for example
while
, then M is not inverse.
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.
is an inverse semigroup if
(i) S and T are inverse semigroups;
(ii)
and
act trivially;
(iii) For each
where
and
, then s and t act trivially on each other.
Proof. We know that
is regular. Since a regular semigroup is inverse if and only if its idem- potents commutes, it suffices to show that idempotents of
commute. If
are idem- potents of
, then
![]()
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
but t and c are idempotents they are act trivially then
![]()
Thus
is inverse. W
5. External Zappa-Szép Products
Let A and B be semigroups, and suppose that we are given functions
and
where
and
satisfying the Zappa-Szép rules
and
Then the set
with the product defined by:
is a semigroup, the external Zappa-Szép product of A and B, which is written as ![]()
If A and B are semigroups that both have zero elements (
and
respectively), and we have in addition to
and
for all
and
the following rule:
(ZS5) ![]()
Then by Proposition [10] we have that S is a semigroup with zero
But from the following example we deduce that
is not a necessary condition:
Example 7. If A and B are semigroups with
and
respectively, acting trivially on each other. Then (ZS5) is not satisfied. However, in the Zappa-Szép product
we have
and
Thus
is zero for ![]()
From the following example we deduce that the zeros 0A and 0B 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
and zero
and B is a semigroup with zero
such that the action of A on B is trivial action
and the action of A on B is
for all
Then Zappa-Szép rules
are satisfied. But (ZS5) is not satisfied since
and
is not a zero for ![]()
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
completes the square
![]()
From the horizontal composition we get
and
as follows:
![]()
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 [13] . In the theory of quantum groups Zappa-Szép product known as the bicrossed (bismash) product see [14] .
6. Internal and External Zappa-Szép Products
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
where for all
and
we have
and
so that the multiplication in P is
Then for each
the subset
is
a subgroup of P isomorphic to
(with identity
). However P cannot be an internal Zappa-Szép pro- duct of subsemigroups Z, N isomorphic to
and
respectively: If
generates N, then the second coordinate of every non-identity element of N is q, and so the second coordinate of any product
with
and
is equal to q.
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
and
respectively such that the following is satisfied
![]()
for all
and ![]()
So if
, the external Zappa-Szép product of A and B, then each A and B are embedded in
Define
by
. We claim
is an injective homomorphism since
and
. Thus τM is a homo-
morphism, also
is injective since
Denote its image by
. Define
by
, then
is also an injective homomorphism. Denote its image by
. Observe that
Thus
. This decomposition is evidently unique. Thus
is the internal Zappa-Szép product of
and
.
・ 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
and
where
are fixed elements of A and B respectively. Note that in a left-zero semi- group A,
and in a right-zero semigroup B,
and we have the following Theorem:
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
, let
and
where
are fixed elements. Then
uniquely and
and
So
is the internal Zappa-Szép product of
and
,
, where
(as left-zero semi- group) and
(as right-zero semigroup).
Conversely, Let
where A is left-zero semigroup and B is right-zero semigroup. Then
for all
and
for all
M is a rectangular band if for all
then
where
and
for unique
and
Now
![]()
Thus M is a rectangular band. W
7. Examples
1) Let
be a Clifford semigroup. Note that
and
Let
the group of integers. Suppose that the action of
on A for each
is as follows:
The action of A on
as follows:
Then the Zappa-Szép multiplication is associative. Thus
the Zappa-Szép product of A and B. The set
of idempotents of M is the empty set, since
if and only if
and
since
or
for all
so
or ![]()
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
since if we define
and
with
we obtain the multi- plication
Then
is the external Zappa-Szép product of A and A. Moreover M is a band if and only if A is a rectangular band, in which case M is a rectangular band.
3) Let
where
and
―Klein 4-group, where
Suppose that the action of G on E is defined by:
and
and
acts trivially. The action of
on
is defined by:
and
Then
is the Zappa-Szép product of E and G. Since every element of M is regular, then M is regular.
is a closed subsemigroup of M, so
is orthodox, but since
is not commutative for example
while
, then M is not inverse.
4) For groups,
is the Zappa-Szép product of subgroups A and B if and only if,
since for any
we have
for unique
and
This implies that
for unique
and
Thus
But this is not true in general for semigroups or monoids. Let
be a commutative monoid with one non-identity idempotent a. Let
be a com- mutative monoid with two idempotents e and f and
Let B act trivially on A and
act trivially
on B and
Then
and
Then
is the internal Zappa-Szép product of
and
. But
, since
so
can not be written as
. Moreover,
so
is not a submonoid of M.
8. Zappa-Szép Products and Nilpotent Groups
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
being contained in the center
of G. Now, let G be a group and let G act on itself by left and right conjugation as follows:
![]()
In the following we show that these actions let us form a Zappa-Szép product
if and only if G is nilpotent group of class at most 2.
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
Thus the multiplication is given by:
![]()
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
for all
For
and
clear they are hold.
![]()
:
:
![]()
since G is nilpotent of class £ 2,then
![]()
Thus
holds.
![]()
:
:
![]()
since G is nilpotent of class 2, then
![]()
Thus
holds. Hence
is the Zappa-Szép product. W
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
holds,
then for all
we have
. Thus
![]()
Therefore
Hence
is central in G. Similarly if
holds. W
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
is ![]()
Proof. Suppose
Since for all
we have
and
Then
(1) ![]()
and
(2) ![]()
Put in (1)
: then
for all
Therefore
So
Put in (2)
: then
for all
Therefore
So
So
since if
Then
W
Lemma 4. If
then P is abelian if and only if G is abelian.
Proof. If G is abelian then G is nilpotent of class 1 if and only if
This implies
and so P is abelian.
If P is abelian then
but
. Thus
if and only if
Hence G is abelian group. In which case
W
Proposition 12. If P is the non-abelian Zappa-Szép product
and G is nilpotent group of class at most 2, then P is nilpotent of class 2.
Proof. We have G is nilpotent group of class £ 2 if and only if for all
we have
that is
the commutator elements are central. Let
be a commutator in P. We prove it is central in P. We have
![]()
Now
![]()
and
![]()
Write
Then
![]()
Since
are commutators, then
![]()
This implies that
Thus
So commutators in P are in the center ( and P is not abelian) so P is nilpotent of class 2. W
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
with left and right con- jugation action of G on it self. Then:
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.
9. Zappa-Szép Products of Semilattices and Groups
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 [11] for example) However, Bernd Billhardt [15] showed how to get around this difficulty in the semidirect product of two inverse semigroups by modifying the definition of semidirect products in the inverse case to obtain what he termed
-semidirect products. The
-semidirect product of inverse semigroups is again inverse. In this Section, we construct from the Zappa-Szép product P of a semilattice E and a group G, an inverse semigroup by constructing an inductive groupoid. We assume the additional axiom for the identity element
we have ![]()
Note that if
for all
then
holds, since by cancellation in the group G.
![]()
We consider the following where E a semilattice and G a group, and subset
of the Zappa-Szép product
:
![]()
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
starts at
finishes at ![]()
・ the inverse of the arrow
is ![]()
・ the identity arrow at e is
and
・ an arrows
and
are composable if and only if
, in which case the composite arrow
is ![]()
Lemma 5. If
then ![]()
Proof. we have
for all
then
Then
W
Lemma 6. Suppose that
then
is a regular element and ![]()
Proof. We have
![]()
and
![]()
Thus
is a regular. W
Proposition 13. If
where E is a semilattice and G is a group then
![]()
with composition defined by
![]()
if
and
acts trivially on E is a groupoid.
Proof.
![]()
We have to prove
& ![]()
![]()
Now
![]()
Since
implies that
this implies that
therefore
![]()
Then
![]()
&
starts at ![]()
But
starts at
and
so ![]()
&
ends at ![]()
But
ends at ![]()
![]()
Thus
is a groupoid. W
Now we introduce an ordering on
. The ordering is giving as follows:
![]()
Lemma 7. The ordering on
defined by
![]()
is transitive.
Proof. We have to prove that if
then
We have
and
and
and
Since
Thus
We conclude
and
Now,
implies that
and
Thus £ is transitive. W
Lemma 8. The ordering on
defined by
![]()
is antisymmetric.
Proof. We have to prove if
and
, then
Now
and
and
and
Thus
and
Thus £ is antisymmetric. W
Proposition 14.
with ordering defined by
![]()
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
is a partial order set. W
Next we prove that
is an ordered groupoid.
Lemma 9. If
, then
for all ![]()
Proof. Suppose that
so that
and
Now, we have
![]()
Thus
![]()
and
![]()
Therefore
Also
![]()
and hence
as required. W
Lemma 10. If
and
such that the composition
and
are defined, then
![]()
for all ![]()
Proof. Suppose that
and
are defined
Then we have
![]()
and
![]()
and we have the following
![]()
where ![]()
![]()
where
Now
![]()
and
![]()
Then
Moreover,
Thus
as required. W
Lemma 11. If
and
is an identity such that
then
is the
restriction of
to ![]()
Proof. Suppose
and
is an identity such that
since
then Also
![]()
Moreover,
and unique by definition.
Thus
is the restriction of
to
W
Proposition 15.
is an inductive groupoid.
Proof. We prove that
and
hold. By Lemma 9 we have
by Lemma 10
holds and by Lemma 11
holds. Since the partially ordered set of identities forms a meet semi-
lattice
. Thus
is an inductive groupoid. W
Theorem 7. If
where E is a semilattice and G is a group, then
![]()
is an inverse semigroup with multiplication defined by
![]()
Proof. Let
since
and
we form
the pseudoproduct
using the greatest lower pound.
![]()
and
![]()
and
![]()
Now, since
then
and so
We have
Then
![]()
Therefore
![]()
Now, we have in the ordering defined on
and
acts trivially on g this
implies that
Then we have
![]()
and
![]()
Therefore
![]()
Thus
is an inverse semigroup. W
We summarize the main results of this paper in the following:
1) We characterize Green’s relations (
and
) of the Zappa-Szép product
of a monoid M and a group G we prove that
in M. And If
such that
and
in G, then
in ![]()
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
and
is the internal Zappa-Szép product of the submonoids
and
where
And 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.
3) We give the necessary and sufficient conditions for the internal Zappa-Szép product
of re- gular subsemigroups A and B to again be regular. We prove that
is regular if and only if
where
and ![]()
4) The Zappa-szep products
of the nilpotent group G with left and right conjugation action of G on it self is abelian if and only if G is abelian, in which case
and if G is non-abelian and hence nilpotent of class 2, then P is also nilpotent of class 2.
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
-semidirect products. The
-semidirect product of inverse semi- groups is again inverse. We construct from the Zappa-Szép product P of a semilattice E and a group G, an inverse semigroup by constructing an inductive groupoid.