Counterexamples in stable semigroups
Liam O’Carroll · Transactions of the American Mathematical Society · 1969
Introduction.A semigroup S is stable if and only if (1) for a and b in S, Sa^Sab implies Sa=Sab, and (2) for c and d in S, cS^dcS implies cS=dcS.A semigroup 5 is pseudo-invertible if and only if some power of every element lies in a subgroup of S.We can adjoin an identity element 1 to a semigroup not having one.If S is an arbitrary semigroup, S1 denotes the semigroup 5u{l} if S does not have an identity element, and denotes S otherwise [2, p. 4].Stable semigroups were first investigated by Koch and Wallace [4], who noted that if a semigroup S is stable then S1 is stable.If S is regular the converse holds.However, the main purpose of this paper is to show that the converse does not hold in general.This possibility was first raised in [1], p. 521, footnote (2).In this paper, Anderson, Hunter and Koch developed the theory of stable semigroups further, and some basic results from [1] and [4] are collected together in Theorem 1.1 below.Let S be a semigroup.S is called weakly stable if S1 is stable.If S is weakly stable but not stable, it is called very weakly stable.A weakly stable semigroup S is called very weakly stable on the left [right] if it fails to satisfy (1) [( 2)] of the definition of a stable semigroup given above.By investigating the structure of the members of a certain class of semigroups very weakly stable on the left, we produce a general method for constructing counterexamples (Theorem 3.5).Pseudo-invertible semigroups, studied by Munn in [5], are used extensively in this paper.We prove that if S is pseudo-invertible then S is weakly stable (Theorem 2.1).Finally we end the paper with a counterexample to a surmise which arises in the course of our work (Theorem 4.4).The equivalences of Green [3], defined for any semigroup S, are as follows: aÓtb[a¿fb, afb] if and only if aS1 = bS^a = S1*», 51a51 = S^S1].¿e = se r\0i, ® = &*0t = ®o£e.For aeS, let L(a)[R(a), J(a)] be the set S^aS1, S^S1].Let La = {xeS:L(x)=L(a)}, with analogous definitions for Ha, Ra, Da, Ja; and let I(a)=J(a)\Ja.