Abstract

This paper is concerned with a certain class of regular semigroups. It is well-known that a regular semigroup in which the set of idempotents satisfies commutativity x₁x₂ = x₂x₁ is an inverse semigroup firstly introduced by V. V. Vagner, and the structure of inverse semigroups was clarified by A. E. Liber, W. D. Munn, G. B. Preston and V. V. Vagner, etc. By a generalized inverse semigroup is meant a regular semigroup in which the set of idempotents satisfies a permutation identity x₁x₂⋯x_n = x_p1x_p2⋯x_pn (where (p₁,p₂,⋯,p_n) is a nontrivial permutation of (1,2,⋯,n)). N. Kimura and the author proved in a previous paper that any band B satisfying a permutation identity satisfies normality x₁x₂x₃x₄ = x_Ix₃x₂x₄. Such a B is called a normal band, and the structure of normal bands was completely determined. In this paper, first a structure theorem for generalized inverse semigroups is established. Next, as a special case, it is proved that a regular semigroup is isomorphic to the spined product (a special subdirect product) of a normal band and a commutative regular semigroup if and only if it satisfies a permutation identity. The problem of classifying all permutation identities on regular semigroups into equivalence classes is afso solved. Finally, some theorems are given to clarify the mutual relations between several conditions on semigroups. In particular, it is proved that an inverse semigroup satisfying a permutation identity is necessarily commutative.

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