R. J. Warne has defined an w<sup>n</sup>-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (I<sup>o</sup>)<sup>n</sup>, where I<sup>o</sup> 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<sup>n</sup>-right cancellative semigroup into an ω<sup>n</sup>-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n ["Lectures in Semigroups," West Virginia Univ., unpublished].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<sup>n</sup>-right cancellative semigroup into an ω<sup>m</sup>-right cancellative semigroup for arbitrary natural numbers n and m.
One of the main results is the following:
Theorem: Let P = (G ,(I<sup>o</sup>)<sup>n</sup> , γ₁,...,γ<sub>n</sub>, w₁,…,w<sub>Ø(n)</sub>) and P<sup>*</sup> = (G ,(I<sup>o</sup>)<sup>n</sup> , α₁,...,α<sub>n</sub>, t₁,…,t<sub>Ø(n)</sub>) be ω<sup>n</sup>-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,z<sub>n</sub> be elements of G<sup>*</sup> and let f be a homomorphism of G into G<sup>*</sup> such that
(1) (Af)<sup>(U<sub>k</sub>g)</sup>C<sub>z<sub>k</sub></sub> = (Aγ<sub>k</sub>f) for A ∈ G where 1 ≤ k ≤ n and
(2) ((z<sub>k+s</sub>)<sup>(U<sub>k</sub>g)</sup>(U<sub>k</sub>g)<sup>(U<sub>k+s</sub>g)</sup>C<sub>z<sub>k</sub></sub> = w<sub>Ø(n-k)+s</sub>f where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements U<sub>k</sub> (1 ≤ k ≤ n) are generators of (I<sup>o</sup>)<sup>n</sup>, xC<sub>z<sub>k</sub></sub> = z<sub>k</sub>xz<sub>k</sub>⁻¹ for x ∈ G<sup>*</sup>, and x<sup>a</sup>,a<sup>b</sup> in G<sup>*</sup> (x ∈ G<sup>*</sup>; a,b ∈ (I<sup>o</sup>)<sup>n</sup> are specified.
Define, for (A,a₁,...,a<sub>n</sub>) ∈ P, (A,a₁,...,a<sub>n</sub>)M = [(Af)(a₁,...,a<sub>n</sub>)h,(a₁,...,a<sub>n</sub>)g] where h is a specified function from (I<sup>o</sup>)<sup>n</sup> into G* and g is a determined endomorphism of (I<sup>o</sup>)<sup>n</sup>. 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<sup>m</sup>-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<sub>S</sub> denote the set of idempotents of S. S is called ω<sup>n</sup>-bisimple if and only if E<sub>S</sub>, under its natural order, is order isomorphic to I x (I<sup>o</sup>)<sup>n</sup> under the reverse lexicographic order n ≥ 1. S is called I-bisimple if and only if E<sub>S</sub>, under its natural order, is order isomorphic to I under the reverse usual order.
Warne has described, modulo groups, the structure of ω<sup>n</sup>-bisimple, ω<sup>n</sup>I-bisirnple and I-bisimple semigroups in ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811], ["ω<sup>n</sup>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<sup>n</sup>-right cancellative semigroups, for the cases (i) S ω<sup>n</sup>-bisimple and S* ω<sup>m</sup>-bisimple and (ii) S I-bisimple or ω<sup>n</sup>I-bisimple and S* I-bisimple or ω<sup>m</sup>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<sup>n</sup>-bisimple, ω<sup>n</sup>I-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/91175 |
| Date | January 1969 |
| Creators | Hogan, John Wesley |
| Contributors | Mathematics |
| Publisher | Virginia Polytechnic Institute |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | iv, 174, [3] leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 20782485 |
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