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Equational bases and nonmodular lattice varieties


Author: Ralph McKenzie
Journal: Trans. Amer. Math. Soc. 174 (1972), 1-43
MSC: Primary 06A20; Secondary 08A15
DOI: https://doi.org/10.1090/S0002-9947-1972-0313141-1
MathSciNet review: 0313141
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Abstract: This paper is focused on equational theories and equationally defined varieties of lattices which are not assumed to be modular. It contains both an elementary introduction to the subject and a survey of open problems and recent work. The concept of a ``splitting'' of the lattice of lattice theories is defined here for the first time in print. These splittings are shown to correspond bi-uniquely with certain finite lattices, called ``splitting lattices". The problems of recognizing whether a given finite lattice is a splitting lattice, whether it can be embedded into a free lattice, and whether a given interval in a free lattice is atomic are shown to be closely related and algorithmically solvable. Finitely generated projective lattices are characterized as being those finitely generated lattices that can be embedded into a free lattice.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0313141-1
Keywords: Lattice varieties, equational bases, irreducible equations, projective lattices, splitting lattices, atomic intervals of a free lattice, algorithms, finitely based equational theories, Jónsson theorem for congruence distributive varieties
Article copyright: © Copyright 1972 American Mathematical Society

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