Abstract
Problem Statement: There are some special classes of semi group namely: regular and eventually regular, abundant, orthodox, quasi-adequate. The objective of this study were to: (i) Define a new class of semi group on a Poset and give related examples (ii) Study and establish conditions that characterized Chain as a regular semi group. Approach: Tests of some of characteristics of semi group like associativity, commutativity, and regular semi group were carried out on this new class. Results: Conditions were obtained that showed it is associative and regular. Conclusion: Hence the results suggest that since Chain is regular, there are many other things we can still do this with class of semi group such as: (i) Whether one can characterize all the Green's equivalences and their starred analogues (ii) Whether one can characterize all the congruencies of the given semi group (iii) Whether one characterize all the subsemigroups of the given semi group.
Highlights
In[2,3], Semi group was established as a non-empty set S with binary operation * such that S is associative on * that is, for all a,b,c ∈S, a*(b*c) = (a*b)*c
Bicyclic semi group: A semi group B = N×N where N is the set of non-negative integers and (m,n)(p,q) = (m-n+t, q-p+t) where {t = max(n,p)} is a Bicyclic semi group
Conditions were established under which Chain is a regular semi group
Summary
If S is a semi group, the element a∈S is said to be regular if there exist b∈S such that aba = a[5]. Bicyclic semi group: A semi group B = N×N where N is the set of non-negative integers and (m,n)(p,q) = (m-n+t, q-p+t) where {t = max(n,p)} is a Bicyclic semi group. E(B) = the set of idempotent of B defined as E(S) = [(m,m)∈Β: m∈Ν}. These have played a very important role in the study of Chain as a semi group. See for example [4,7]
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