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U.W. Bangor - School of Informatics
Mathematics Preprints 2001

Semigroup and Automata Theory


01.13 : LAWSON, M.V. & MATTHEWS, J.

The universal group of an inverse semigroup

Abstract:

We define the universal group of an inverse semigroup with zero and examine its properties.
In the case of inverse semigroups without zero we recapture the familiar minimum group congruence.

- to be replaced by an up-dated version.


01.14 : LAWSON, M.V., MATTHEWS, J. & PORTER, T.

The homotopy theory of inverse semigroups

Abstract:

We show that abstract homotopy theory can be used to define a suitable notion of homotopy equivalence for inverse semigroups.
As an application of our theory, we prove a theorem for inverse semigroup homomorphisms which is the exact counterpart of the well-known result in topology which states that every continuous function can be factorised into a homotopy equivalence followed by a fibration.
We show that this factorisation is isomorphic to the one constructed by Steinberg in his `Fibration Theorem', originally proved using a generalisation of Tilson's derived category.

Published in:

IJAC 12 (2002) 755-790.

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gzipped PostScript file: 01_14.ps.gz


01.31 : HINES, P.

A short note on coherence and self-similarity

Abstract:

Mac Lane's original introduction to the theory of monoidal categories presented a short argument, due to J. Isbell, of why the concept of associativity up to isomorphism is needed for a reasonable conception of a monoidal tensor. This argument was based on the properties of a distinguished object D in a category with product, satisfying D = D x D. In the following paper, we demonstrate that a slight modification of this property allows us to construct elements of End(D) that have similar properties to associative isomorphisms in a monoidal category, and show how these can be used to construct what can reasonably be considered to be a weakening of the associativity of a strict monoidal category.

Published in:

J Pure and Applied Algebra (to appear).

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gzipped PostScript file: 01_31.ps.gz


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