1. Introduction
Let A and B be arbitrary monoids. In [2] , Theorem 2.2, Howie and Ruskuc defined a presentation for the (restricted) wreath product of A and B. Also, in [3] , Theorem 7.1, it has been showed that the wreath product of semigroups satisfies the periodicity when these semigroups are periodic. In [1] , a new derivation for wreath product of monoids A and B has been recently defined which will be dented by
here (in [1] , it has been denoted by
but the author prefers her the symbol
instead of
to distinguish this new type of extension from the known symbol for general product (Zappa-Szép product
)). Also, again in [1] , by proving the existence a presentation of this wreath product, it has been given necessary and sufficient conditions for
to be regular and periodic, and some finite and infinite applications about it are denoted. In this paper, we give some algebraic properties of the new wreath product in terms of the algebraic properties of the monoids themselves. More specifically, we present the Green’s relation
and
(in Section 2), and also prove the conditions on it to be left cancellative, orthodox and left (right) inverse.
We recall the fundamentals of the construction of
which will be needed to form our results. We note that this product is based on the wreath product and we may refer to ( [2] [4] [5] [6] [7] ) for the details of wreath products. The Cartesian product of B copies of the monoid A is denoted by
, while the corresponding direct product is denoted by
, similar definition for
. One may think that
and
are the sets of all functions having finite support, that is to say, having the property that
for all but finitely many b in B and
for all but finitely many a in A. The restricted wreath product of the monoid A by the monoid B is the set
with the multiplication defined by
(1)
where,
is given by
(2)
Dually the restricted wreath product of the monoid B by the monoid A is the set
with the multiplication defined by
(3)
where,
is given by
(4)
Now for
,
, let us define
After that the new derivation for the wreath product of A and B, denoted by
, is the set
with the multiplication
(5)
where,
and
are defined by
(6)
and
(7)
In fact,
is a monoid with the identity
, where
and
are defined by
(8)
respectively, for all
and
.
2. Green’s Relations on the Product
In the light of Green’s relations, it is well known that one may prove some computational results (for example, the minimal number of generators etc.) on the monoid structure (which will be kept for a future work and so not investigated in here). Hence, in this section, we only characterize Green’s relations
and
(cf. [8] [9] ) for the product
.
Proposition 2.1 Let
be the new derivation of wreath product of a monoid A by a monoid B. Then
1)
in
implies that
in
, and
in
,
2)
in
implies that
in
, and
in
.
Proof. 1) Suppose that
in
. So there exist
such that
(9)
(10)
These two equations can also be written as
(11)
(12)
Hence, by the equality of components, we obtain
(13)
(14)
It follows that
in
while
in
.
Similar proof can be applied for 2). Hence the result. +
Theorem 2.2 Assume that the product
is obtained by a monoid A and a group B. Then
Proof. By Proposition 2.1,
implies the existence of
and
.
To prove the converse, let us suppose that
in
and
in
. In fact,
in
gives that there exist
and
in
such that
Also,
in
implies that there exist
,
in
such that
To show that
in
, we have to find two elements
and
such that these must satisfy
(15)
(16)
From these above, we obtain
(17)
(18)
Since
is a group (because B is a group), we have
(19)
(20)
Therefore, we set
and
. Hence
With a similar way, one can also show that
Hence,
, as required. +
3. Some Algebraic Properties on
In this section, we will illustrate some algebraic properties of the new wreath product
in terms of the algebraic properties of the monoids A and B themselves. The following Theorem characterize when new wreath product
is a group.
Theorem 3.1 The new derivation of wreath product
of monoids A and B is a group if and only if both A and B are groups.
Proof. Suppose A and B are both, groups, then
is a monoid with identity
where
and
are defined by
(21)
Now, let
. Define
Then
Since
, and
. Hence
is a right inverse for
. Also
Since
, and
. Hence,
is a left inverse for
, therefore, M is a group.
Conversely, assume that
is a group, let
so
and
Then
,
,
, and
. Since
and
.
Hence,
, therefore,
is a group and hence B is a group. Similarly we get
, if we suppose that
, therefore
is a group and hence A is a group. +
We first recall that a semigroup S is called left-cancellative if
and right-cancellative if
, for all
(cf. [8] ). A semigroup is cancellative if it is left-cancellative and right-cancellative.
Theorem 3.2 A and B are cancellative monoids if and only if
is cancellative monoid.
Proof. Assume that A and B are left cancellative monoids. Suppose
where
,
and
. Therefore
[Since B is left cancellative]
Also
[Since A is left cancellative]
[Since B is left cancellative]
As a result,
is actually a right cancellative monoid. In fact, one may prove with a similar way for left cancellative. Hence
is cancellative.
On the other hand, the converse part of the proof is clear.
Hence the result. +
In [10] , the question of orthodox wreath products of monoids has been explained. After that, in [11] , it has been investigated the orthodox wreath products of semigroups without unity. In this part, we will give necessary and sufficient conditions for
to be orthodox, where A and B are any monoids.
Recall that the semigroup S is called orthodox if the set of idempotents
is a subsemigroup of S. An orthodox semigroup S is left (respectively, right) inverse if
(respectively,
) for every
. For more details reader refer [12] [13] . From [ [10] , Lemma 3.1],
is an orthodox or left or right inverse semigroup if and only if A has the same property. In the reference [1] , the necessary and sufficient conditions for the new derivation of wreath products to be regular have been defined. In here, we give sufficient conditions for it to be orthodox as in the following theorem.
Theorem 3.3 If
is an orthodox monoid or left (right) inverse, then each of A and B has the same property.
Proof. Applying [ [1] , Theorem 5.2], we see that A and B are regular. It remains to prove that the set of idempotents of A and B closed under multiplication defined on A and B respectively. Let
and
. Then
,
,
and k2 = k. Since
is an orthodox monoid, then for an element
, we have
Furthermore
Similar, calculation shows that
, for every
. Hence the set of idempotents of
and
are subsemigroups. The result follows from [ [10] , Lemma 3.1].
Now let us suppose that
is the left inverse. Then, for any element
, where
and
, we certainly have
and
Hence,
for every
. Similar, calculation shows that
for every
. We thus conclude that A and B are actually left inverses.
The same proof can be applied to show right inverse case as well. +
Note 3.4 1) The other inclusion of Theorem 3.3 is left for future work. Following Caito [10] , who determined necessary and sufficient conditions for the (restricted) wreath product to be orthodox, to be left inverse and to be right inverse, respectively.
2) There is also a particular class of regular monoids, namely coregular monoids [14] . An element of a monoid S is called coregular if there is a
such that
as well as the monoid S is called coregular if each element of it is coregular cf. [15] . In fact, the coregularity and its properties over the new type of wreath product are left as an open problem for the future studies.
4. Conclusion
In this paper, the author investigated some specific theories such as Green’s relations, left cancellative, orthodox, left (right) inverse etc. over new type of wreath products over monoids. Of course, there are still so many different properties that can be checked on this important product. On the other hand, in Note 3.4, we indicated some problems for future studies.