Semigroup of k-bi-Ideals of a Semiring with Semilattice Additive Reduct

Abstract

We associate a semigroup B(S) to every semiring S with semilattice additive reduct, namely the semigroup of all k-bi-ideals of S; and such semirings S have been characterized by this associated semigroup B(S). A semiring S is k-regular if and only if B(S) is a regular semigroup. For the left k-Clifford semirings S, B(S) is a left normal band; and consequently, B(S) is a semilattice if S is a k-Clifford semiring. Also we show that the set Bm(S) of all minimal k-bi-ideals of S forms a rectangular band and Bm(S) is a bi-ideal of the semigroup B(S).

Keywords

• k-bi-ideals
• minimal k-bi-ideals
• k-regular
• k-Clifford
• left normal band