The Source of Semiprimeness of Semigroups

In this study, we define new semigroup structures using the set S S = a ∈ S | a S a = 0 which is called the source of semiprimeness for a semigroup S with zero element. S S − idempotent semigroup, S S − regular semigroup, S S − reduced semigroup, and S S − nonzero divisor semigroup which are generalizations of idempotent, regular, reduced, and nonzero divisor semigroups in semigroup theory are investigated, and their basic properties are determined. In addition, we adapt some well-known results in semigroup theory to these new semigroups.

FINITE REGULAR BANDS ARE FINITELY RELATED

An algebra A is said to be finitely related if the clone Clo(A) of its term operations is determined by a finite set of finitary relations. We prove that each finite idempotent semigroup satisfying the identity xyxzx≈xyzx is finitely related.

Semigroup-graded rings with finite support

Let S be a semigroup and let be an S-graded ring. Rs = 0 for all but finitely many elements s ∈ S1, then R is said to have finite support. In this paper we concern ourselves with the question of whether a graded ring R with finite support inherits a given ring theoretic property from the homogeneous subrings Re corresponding to idempotent semigroup elements e.