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

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R. J. Warne has defined an w^{n}-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (I^{o})^{n}, where I^{o} 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 ω^{n}-right cancellative semigroup into an ω^{n}-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n ["Lectures in Semigroups," West Virginia Univ., unpublished].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 ω^{n}-right cancellative semigroup into an ω^{m}-right cancellative semigroup for arbitrary natural numbers n and m.

One of the main results is the following:

Theorem: Let P = (G ,(I^{o})^{n} , γ₁,...,γ_{n}, w₁,…,w_{Ø(n)}) and P^{} = (G ,(I^{o})^{n} , α₁,...,α_{n}, t₁,…,t_{Ø(n)}) be ω^{n}-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,z_{n} be elements of G^{} and let f be a homomorphism of G into G^{} such that
(1) (Af)^{(Ukg)}C_{zk} = (Aγ_{k}f) for A ∈ G where 1 ≤ k ≤ n and

(2) ((z_{k+s})^{(Ukg)}(U_{k}g)^{(Uk+sg)}C_{zk} = w_{Ø(n-k)+s}f where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements U_{k} (1 ≤ k ≤ n) are generators of (I^{o})^{n}, xC_{zk} = z_{k}xz_{k}⁻¹ for x ∈ G^{}, and x^{a},a^{b} in G^{} (x ∈ G^{}; a,b ∈ (I^{o})^{n} are specified.

Define, for (A,a₁,...,a_{n}) ∈ P, (A,a₁,...,a_{n})M = [(Af)(a₁,...,a_{n})h,(a₁,...,a_{n})g] where h is a specified function from (I^{o})^{n} into G* and g is a determined endomorphism of (I^{o})^{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 ω^{m}-right cancellative semigroup with m < n and m > n.

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

Warne has described, modulo groups, the structure of ω^{n}-bisimple, ω^{n}I-bisirnple and I-bisimple semigroups in ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811], ["ω^{n}I-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 ω^{n}-right cancellative semigroups, for the cases (i) S ω^{n}-bisimple and S* ω^{m}-bisimple and (ii) S I-bisimple or ω^{n}I-bisimple and S* I-bisimple or ω^{m}I-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 ω^{n}-bisimple, ω^{n}I-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date.