## Graduate Theses and Dissertations

12-2012

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics (PhD)

#### Department

Mathematical Sciences

Boris M. Schein

Mark Arnold

Mark Johnson

#### Keywords

Pure sciences, Difunctional relations, Group theory and generalizations, Inverse, Inverse semigroups, Symmetry

#### Abstract

A semigroup S is called inverse if for each s in S, there exists a unique t in S such that sts = s and tst = t. A relation σ contained in X x Y is called full if for all x in X and y in Y there exist x' in X and y' in Y such that (x, y') and (x', y) are in σ, and is called difunctional if σ satisfies the equation σ σ-1 σ = σ. Inverse semigroups were introduced by Wagner and Preston in 1952 and 1954, respectively, and difunctional relations were introduced by Riguet in 1948. Schein showed in 1965 that every inverse semigroup is isomorphic to an inverse semigroup of full difunctional relations and proposed the following question: given an inverse semigroup S, can we describe all of its representations by full difunctional relations? We demonstrate that each such representation may be constructed using only S itself.

It so happens that the full difunctional relations on a set X are essentially the bijections among its quotients. This observation invites us to consider Schein's question as fundamentally a problem of symmetry, as we explain. By Cayley's Theorem, groups are naturally represented by permutations, and more generally, every permutation representation of a group can be constructed using representations induced by its subgroups. Analogously, by the Wagner-Preston Theorem, inverse semigroups are naturally represented by one-to-one partial mappings, and every representation of an inverse semigroup can be constructed using representations induced by certain of its inverse subsemigroups. From a universal algebraic point of view the permutations and one-to-one partial functions on a set X are the automorphisms (global symmetries) of X and the isomorphisms among subsets (local symmetries) of X, respectively. Inspired by the interpretation of difunctional relations as isomorphisms among quotients, or colocal symmetries, we introduce a class of partial algebras which we call inverse magmoids. We then show that these algebras include all inverse semigroups and groupoids and play a role among difunctional relations analogous to that played by groups among permutations and of inverse semigroups among one-to-one partial functions.

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