ON SLA-IDEALS
Ladislav Satko, Otokar Grošek · Mathematica Slovaca · 1995
The aim of this paper is to study left A-ideals which are at th e same tim e semigroups, and to give an outline of th e extent to which this notion is useful. The notion has a ver y close relation to known notions such as quasi-z eros, mild-ideals, and directed groups. Th e re are also some connections with left-simple semigroups having no idempotents. The main result is a description of minimal semigroup left A-ideals in the commutative case. Left A-ideals appear in various areas of mathematics and unify several no tions. They are a generalization of left ideals in semigroups because any left ideal is at the same time a left A-ideal of a semigroup. We will deal with left A-ideals which are also subsemigroups of a given semigroup. In the theory of semigroups one question arises naturally: Does there exist a minimal left A-ideal in the class of all left A-ideals which are at the same time subsemigroups of a given semigroup? We give, in Theorem 12, a complete answer in the commutative case. The situation in the noncommutative case is discussed at the end of the paper. There exists also a ver y close relation to directed groups. First we briefl y recall some notions. DEFINITION 1. ((6)) A nonempt y subset GL of a semigroup S is called a left A-ideal of S (LA-ideal) if SGL H GL 7-= 0 for any s G S. A nonempty subset GR of a semigroup S is called a right A-ideal of S (RA-ideal) if GRS n GR 7^ 0 for any s G S. B y two-sided A-ideal, or simpl y A-ideal, we mean ct subset of S which is both a left and a right A-ideal of S.