The Journal of Symbolic Logic

Research Article

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

Marcel Jackson

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

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.

(Received January 09 2006)

Key words and phrases

  • Quasi-variety;
  • universal Horn class;
  • flat algebra;
  • flat semilattice;
  • membership problems;
  • undecidability;
  • the finite basis problem;
  • flat extension of a group;
  • inverse semigroup;
  • Clifford semigroup;
  • Brandt semigroup;
  • Q-universal;
  • agreeable semigroup