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⋯xn= xp1xp2⋯xpn (where
(p1,p2,⋯,pn) 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= xIx3x2x4. 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.
