Applied Mathematics
Vol.06 No.06(2015), Article ID:56974,21 pages
10.4236/am.2015.66096
Zappa-Szép Products of Semigroups
Suha Wazzan
Mathematics Department, King Abdulaziz University, Jeddah, Saudi Arabia
Email: swazzan@kau.edu.sa
Copyright © 2015 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 12 May 2015; accepted 6 June 2015; published 9 June 2015
ABSTRACT
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 l-semidirect product to the Zappa-Szép product of a semilattice E and a group G by constructing an inductive groupoid.
Keywords:
Inverse Semigroups, Groups, Semilattice, Rectangular Band, Semidiret, Regular, Enlargement, Inductive Groupoid
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.
References
- Zappa, G. (1940) Sulla construzione dei gruppi prodotto di due dadi sottogruppi permutabili tra loro. Atti del secondo congresso dell’Unione Matematica Italiana, Cremonese, Roma, 119-125.
- Lawson, M. (2008) A Correspondence between a Class of Monoids and Self-Similar Group Actions I. Semigroup Forum, 76, 489-517. http://dx.doi.org/10.1007/s00233-008-9052-x
- Brownlowe, N., Ramagge, J., Robertson, D. and Whittaker, M. (2014) Zappa-Szép Products of Semigroups and Their C*-Algebras. Journal of Functional Analysis, 266, 3937-3967. http://dx.doi.org/10.1016/j.jfa.2013.12.025
- Gould, V. and Zenab, R. (2013) Semigroups with Inverse Skeletons and Zappa-Szép Products. CGASA, 1, 59-89.
- Gilbert, N.D. and Wazzan, S. (2008) Zappa-Szép Products of Bands and Groups. Semigroup Forum, 77, 438-455. http://dx.doi.org/10.1007/s00233-008-9065-5
- Wazzan, S.A. (2008) The Zappa-Szép Product of Semigroups. PhD Thesis, Heriot-Watt University, Edinburgh.
- Lawson, M.V. (1998) Inverse Semigroups. World Scientific, Singapore City.
- Lawson, M.V. (1996) Enlargements of Regular Semigroups. Proceedings of the Edinburgh Mathematical Society (Series 2), 39, 425-460. http://dx.doi.org/10.1017/S001309150002321X
- Lawson, M.V. and Marki, L. (2000) Enlargement and Covering by Rees Matrix Semigroups. Springer-Verlag, Berlin, 191-195.
- Kunze, M. (1983) Zappa Products. Acta Mathematica Hungarica, 41, 225-239. http://dx.doi.org/10.1007/BF01961311
- Nico, W.R. (1983) On the Regularity of Semidirect Products. Journal of Algebra, 80, 29-36. http://dx.doi.org/10.1016/0021-8693(83)90015-7
- Hall, T.E. (1982) Some Properties of local Subsemigroups Inherited by Larger Subsemigroups. Semigroup Forum, 25, 35-49. http://dx.doi.org/10.1007/BF02573586
- Fiedorowicz, Z. and Loday, J.L. (1991) Crossed Simplicial Groups and Their Associated Homology. Transactions of the American Mathematical Society, 326, 57-87. http://dx.doi.org/10.1090/S0002-9947-1991-0998125-4
- Kassel, C. (1995) Quantum Groups. Graduate Texts in Mathematics, No. 155, Springer, New York.
- Billhardt, B. (1992) On a Wreath Product Embedding and Idempotent Pure Congruences on Inverse Semigroups. Semigroup Forum, 45, 45-54. http://dx.doi.org/10.1007/BF03025748