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Varieties of equality structures

Fearnley-Sander, D. and Stokes, T. (2003) Varieties of equality structures. International Journal of Algebra and Computation, 13 (04). pp. 463-480.

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We consider universal algebras which are monoids and which have a binary operation we call internalized equality, satisfying some natural conditions. We show that the class of such E-structures has a characterization in terms of a distinguished submonoid which is a semilattice. Some important varieties (and variety-like classes) of E-structures are considered, including E-semilattices (which we represent in terms of topological spaces), E-rings (which we show are equivalent to rings with a generalized interior operation), E-quantales (where internalized equalities on a fixed quantale in which 1 is the largest element are shown to correspond to sublocales of the quantale), and EI-structures (in which an internalized inequality is defined and interacts in a natural way with the equality operation).

Item Type: Journal Article
Murdoch Affiliation: School of Mathematical and Physical Sciences
Publisher: World Scientific Publishing
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