A new monoid construction under crossed products
© Emin et al.; licensee Springer. 2013
Received: 12 December 2012
Accepted: 26 April 2013
Published: 15 May 2013
In this paper we define a new monoid construction under crossed products for given monoids. We also present a generating set and a relator set for this product. Finally, we give the necessary and sufficient conditions for the regularity of it.
MSC:05C10, 05C12, 05C25, 20E22, 20M05.
1 Introduction and preliminaries
In , some conditions for the regularity of the semi-direct product are given. Moreover, in , a new monoid construction under semi-direct product and Schützenberger product for any two monoids is defined, and its regularity is examined. Also, in , necessary and sufficient conditions for this new product to be strongly π-inverse are determined. The regularity and π-inverse property for the Schützenberger product are studied in . By using similar methods as in these above papers, the purpose of this paper is to define a new monoid construction under a crossed product and to give its regularity.
for all , . The crossed system is called normalized if . The map is called weak action and is called an α-cocycle.
If is a normalized crossed system, then we have and by .
for all , . Then is a monoid with unit if and only if is a normalized crossed system. In this case, the monoid is called a crossed product of A and B associated to the crossed system . The reader is referred to  and  for more details on this material.
It is known that is a monoid with identity , where ∅ is an empty set (see ).
2 A new monoid construction
In this section, as one of the main results of the paper, we define a new monoid construction under a crossed product and the Schützenberger product by considering the definitions given in the above section. In order to do that, firstly we give the definition of this new product and then we define its presentation.
on the set , where and are given in Definition 1.
Let us denote this new product by . Then, by the above argument, we say that is a monoid with the identity .
By the following remark, we explain why this new product is worked on in this paper.
Remark 1 In [8–13] the authors give some new results about the p-Cockcroft property of some extensions. So, by using these papers, one can also work on this subject by using this new product. So, one can give some new efficient (equivalently, p-Cockcroft) presentation examples. By the way, one can also do further algebraic works on this new product. For instance, in this paper, we give necessary and sufficient conditions for this new product to be regular.
3 Presentation of
Let us consider Remark 1. In order to do such algebraic work, we need to define the presentation of this new product. So, in the following theorem, we give a presentation of as one of the main results of this paper.
where is the word on X.
Thus these above arguments say that ψ induces an epimorphism from the monoid defined by (3)-(7), say M, onto .
for any word .
Now, let us take and for some . If , then, by the equality of these components, we deduce that in A, in B and . Relations (3) and (4) imply that and hold in M. So that holds. Thus is injective. □
4 Regularity of
Let A and B be monoids. As depicted in Remark 1, one can work on this new product to show some algebraic properties. To this end, in this section we define the necessary and sufficient conditions for to be regular.
For an element a in a monoid M, let us take for the set of inverses of a in M, that is, . Hence M is regular if and only if, for all , the set is not equal to the empty set.
Let us consider the notations given in Definition 3. Then we have the following theorem as a final main result of this paper.
Theorem 2 Let A and B be any monoids. The product is regular if and only if A is a regular monoid and B is a group.
and, in particular, and , we get . This says that B is a group.
where , and for some such that . Hence the result. □
Dedicated to Professor Hari M Srivastava.
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