Let A and B be arbitrary monoids. In  , Theorem 2.2, Howie and Ruskuc defined a presentation for the (restricted) wreath product of A and B. Also, in  , Theorem 7.1, it has been showed that the wreath product of semigroups satisfies the periodicity when these semigroups are periodic. In  , a new derivation for wreath product of monoids A and B has been recently defined which will be dented by here (in  , 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  , 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 (      ) 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
where, is given by
Dually the restricted wreath product of the monoid B by the monoid A is the set with the multiplication defined by
where, is given by
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
where, and are defined by
In fact, is a monoid with the identity , where and are defined by
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.   ) 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
These two equations can also be written as
Hence, by the equality of components, we obtain
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
From these above, we obtain
Since is a group (because B is a group), we have
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
Now, let . Define
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
Then , , , and . Since
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.  ). 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]
[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  , the question of orthodox wreath products of monoids has been explained. After that, in  , 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   . From [  , Lemma 3.1], is an orthodox or left or right inverse semigroup if and only if A has the same property. In the reference  , 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 [  , 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
Similar, calculation shows that , for every . Hence the set of idempotents of and are subsemigroups. The result follows from [  , Lemma 3.1].
Now let us suppose that is the left inverse. Then, for any element , where and , we certainly have
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  , 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  . 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.  . In fact, the coregularity and its properties over the new type of wreath product are left as an open problem for the future studies.
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.
 Robertson, E.F., Ruskuc, N. and Thomson, M.R. (2002) On Finite Generation and Other Finiteness Conditions for Wreath Products of Semigroups. Communications in Algebra, 30, 3851-3873.