The CONGRUENCE Η* ON SEMIGROUPS

Abstract

In this paper we define a congruence η * on semigroups. For the finite semigroups S, η * is the smallest congruence relation such that S η * is a nilpotent semigroup (in the sense of Mal'cev). In order to study the congruence relation η * on finite semigroups, we define a CSdiagonal finite regular Rees matrix semigroup. We prove that, if S is a CS-diagonal finite regular Rees matrix semigroup then S η * is inverse. Also, if S is a completely regular finite semigroup, then S η * is a Clifford semigroup.We show that, for every non-null principal factor A B of S, there is a special principal factor C D such that every element of A ∖ B is η *equivalent with some element of C ∖D. We call the principal factor C D, the η * -root of A B. All η * -roots are CS-diagonal. If certain elements of S act in the special way on the R-classes of a CS-diagonal principal factor then it is not an η * -root. Some of these results are also expressed in terms of pseudovarieties of semigroups.

Nilpotency and strong nilpotency for finite semigroups

“…The smallest such n is called the nilpotency class of S. Clearly, null semigroups are nilpotent in the sense of Mal 'cev. A pseudovariety of semigroups is a class of finite semigroups closed under taking subsemigroups, homomorphic images and finite direct products. The finite nilpotent semigroups constitute a pseudovariety which is denoted by MN [18]. In [5], the rank of the pseudovariety MN and some classes defined by several of the variants of Mal'cev nilpotent semigroups are investigated and they are compared.…”

The rank of variants of nilpotent pseudovarieties

“…Recall that, the vertices of N S are the elements of S and there is an edge between x and y, if the subsemigroup generated by x and y is not nilpotent. Recall from [23] that the collection of finite semigroups whose upper non-nilpotent graphs have no edges is a pseudovariety which is denoted EUNNG. Obviously, the rank of the pseudovariety EUNNG is equal to 2.…”