Homomorphisms of wn-right cancellative, wn-bisimple, and wnI-bisimple semigroups

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1969

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Virginia Polytechnic Institute

Abstract

R. J. Warne has defined an wn-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (Io)n, where Io is the set of non-negative integers and n is a natural number, under the reverse lexicographic order. Warne has described, modulo groups, the structure of such semigroups ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811]. He has used this structure and the theory of right cancellative semigroups having identity on which Green's relation J: is a congruence to describe the homomorphisms of an ωn-right cancellative semigroup into an ωn-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n ["Lectures in Semigroups," West Virginia Univ., unpublished].

We have described, modulo groups, the homomorphisms of an ωn-right cancellative semigroup into an ωm-right cancellative semigroup for arbitrary natural numbers n and m.

One of the main results is the following:

Theorem: Let P = (G ,(Io)n , γ₁,...,γn, w₁,…,wØ(n)) and P = (G ,(Io)n , α₁,...,αn, t₁,…,tØ(n)) be ωn-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,zn be elements of G and let f be a homomorphism of G into G such that (1) (Af)(Ukg)Czk = (Aγkf) for A ∈ G where 1 ≤ k ≤ n and
(2) ((zk+s)(Ukg)(Ukg)(Uk+sg)Czk = wØ(n-k)+sf where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements Uk (1 ≤ k ≤ n) are generators of (Io)n, xCzk = zkxzk⁻¹ for x ∈ G, and xa,ab in G (x ∈ G; a,b ∈ (Io)n are specified.

Define, for (A,a₁,...,an) ∈ P, (A,a₁,...,an)M = [(Af)(a₁,...,an)h,(a₁,...,an)g] where h is a specified function from (Io)n into G* and g is a determined endomorphism of (Io)n. Then, M is a homomorphism of P into P* and every homomorphism of P into P* is obtained in this fashion. M is an isomorphism if and only if f and g are isomorphisms. M is onto when g is the identity and f is onto.

Results similar to this theorem have been obtained when P* is an ωm-right cancellative semigroup with m < n and m > n.

Let I be the set of integers. Let S be a bisimple semigroup and let ES denote the set of idempotents of S. S is called ωn-bisimple if and only if ES, under its natural order, is order isomorphic to I x (Io)n under the reverse lexicographic order n ≥ 1. S is called I-bisimple if and only if ES, under its natural order, is order isomorphic to I under the reverse usual order.

Warne has described, modulo groups, the structure of ωn-bisimple, ωnI-bisirnple and I-bisimple semigroups in ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811], ["ωnI-bisimple Semigroups," to appear], and ["I-bisimple Semigroups," Trans. Amer. Math. Soc., Vol. 130 (1968), pp. 367-386] respectively.

We have described the homomorphisms of S into S* , by use of the homomorphism theory of ωn-right cancellative semigroups, for the cases (i) S ωn-bisimple and S* ωm-bisimple and (ii) S I-bisimple or ωnI-bisimple and S* I-bisimple or ωmI-bisimple where m and n are natural numbers. The homomorphisms of S onto S* are specified for cases (i) and (ii). Warne has determined the homomorphisms of S onto S* in certain of these cases as he studied the extensions and the congruences of ωn-bisimple, ωnI-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date.

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