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Flat algebras and the translation of universal Horn logic to equational logic

Published online by Cambridge University Press:  12 March 2014

Marcel Jackson
Marcel Jackson*
Affiliation:
La Trobe University, Department of Mathematics, Victoria 3086, Australia, E-mail: m.g.jackson@latrobe.edu.au
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Abstract

We describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, partial algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities is shown to be equivalent to the finite identity basis problem for the finite members of a finiteiy based variety with definable principal congruences.

Keywords

Quasi-varietyuniversal Horn classflat algebraflat semilatticemembership problemsundecidabilitythe finite basis problemflat extension of a groupinverse semigroupClifford semigroupBrandt semigroupQ-universalagreeable semigroup
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 1 , March 2008 , pp. 90 - 128
Copyright
Copyright © Association for Symbolic Logic 2008

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