Vol. 60, No. 1, 1975

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On semigroups in which x = xyx = xzx if and only if x = xyzx

Zensiro Goseki

Vol. 60 (1975), No. 1, 103–110
Abstract

A semigroup S will be called quasi-rectangular if the set of idempotents of S is non-empty and a rectangular band ideal of S. The theorems of this note prove in part that the following are equivalent. (1) S is a semilattice of semigroups each of which is either idempotent free or quasi-rectangular. (2) Every 𝒥 -class of S is either idempotent free or a rectangular subband of S. (3) Every 𝒟-class of S is either idempotent free or a rectangular subband of S. (4) S is a semigroup in which for any x,y,z S, x = xyx = xzx if and only if x = xyzx.

Mathematical Subject Classification 2000
Primary: 20M10
Milestones
Received: 17 May 1974
Published: 1 September 1975
Authors
Zensiro Goseki