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Equality algebras

Published online by Cambridge University Press:  17 April 2009

Desmond Fearnley-Sander  and
Tim Stokes
Desmond Fearnley-Sander
Affiliation:
Department of Mathematics, University of Tasmania, Hobart Tas 7000
Tim Stokes
Affiliation:
Department of Mathematics and Statistics, Murdoch University, Murdoch Wa 6150
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Abstract

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As part of an attempt to capture abstractly the most fundamental properties of algebraic reasoning involving equality, we introduce the notion of an equality algebra. It is a universal algebra A endowed with a binary function =iA × A → L, where L is a meet-semilattice with top element 1, called internalised equality, and satisfying, for all x, y ∈ A,

1. (x =ix) = 1; and

2. (x =iy)f(x) = (x =iy)f(y), where f is any function A → L derived from the Operations on A, the semilattice operations, and = i.

We charecterise internalised equalities in terms of finetly many identities, give examples, and show that all are equivalent to internalised equalities defined in terms of congruences on the underlying algebra. In the special case in which A is an Abelian group or ring, the internalised equality is shown to be equivalent to the dual of a norm-like mapping taking values in semilattice.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 177 - 191
Copyright
Copyright © Australian Mathematical Society 1997

References

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