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

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 (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 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into an ωⁿ-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n ["Lectures in Semigroups," West Virginia Univ., unpublished].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 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 ωⁿ-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)^(U_kg)C_zk = (Aγ_kf) for A ∈ G where 1 ≤ k ≤ n and
(2) ((z_k+s)^(U_kg)(U_kg)^(U_k+sg)C_zk = w_Ø(n-k)+sf 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_kxz_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 ωⁿ-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 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* ω^m-bisimple and (ii) S I-bisimple or ωⁿI-bisimple and S* I-bisimple or ω^mI-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.