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Flat algebras and the translation of universal Horn logic to equational logic. (English) Zbl 1141.03006

This paper has arisen out of a proof in [M. Jackson and T. Stokes, Int. J. Algebra Comput. 16, No. 6, 1131–1159 (2006; Zbl 1117.08003)], showing that the variety of Clifford semigroups whose natural order is a meet semilattice order has undecidable equational theory. The proof is by an interpretation of semigroup quasi-identities as identities in the enlarged signature involving the semilattice \(\wedge .\) In this paper the author is going to describe how this result is a part of a completely general translation of universal Horn classes (henceforth, uH classes) of partial algebras into varieties of conventional (that is, not partial) algebras. More precisely, he finds an isomorphism from the lattice of all uH classes of a given similarity type to the lattice of subvarieties of a particular variety. This isomorphism translates the main global properties of the uH class to the corresponding variety properties. The constructed varieties are hereditarily simple, semisimple and have definable principal congruences. So he has a very precise interpretation of uH logic inside the equational logic of some apparently quite well-behaved varieties. Furthermore, the constructed varieties are generated by flat algebras, which attracted increasing interest since McKenzie’s landmark articles on undecidability, residual bounds and axiomatisability.

MSC:

03C05 Equational classes, universal algebra in model theory
08A55 Partial algebras
08B15 Lattices of varieties
08B26 Subdirect products and subdirect irreducibility
08C15 Quasivarieties
20A15 Applications of logic to group theory
20E10 Quasivarieties and varieties of groups
20M18 Inverse semigroups

Citations:

Zbl 1117.08003
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