Varieties of Lattices

AbstractThe homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterise varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence n-permutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.

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Cut elimination and word problems for varieties of lattices

Algebra Universalis ◽

10.1007/bf02483891 ◽

1981 ◽

Vol 12 (1) ◽

pp. 290-321 ◽

Cited By ~ 10

Author(s):

Jürgen Schulte Mönting

Keyword(s):

Word Problems ◽

Cut Elimination ◽

Varieties Of Lattices

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A Remark on Varieties of Lattices and Semigroups

Proceedings of the American Mathematical Society ◽

10.2307/2037012 ◽

1969 ◽

Vol 21 (2) ◽

pp. 394 ◽

Cited By ~ 1

Author(s):

Richard A. Dean ◽

Trevor Evans

Keyword(s):

Varieties Of Lattices

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Mapping Extension of an Algebra

Algebra Colloquium ◽

10.1142/s1005386709000455 ◽

2009 ◽

Vol 16 (03) ◽

pp. 479-494

Author(s):

Adam W. Marczak ◽

Jerzy Płonka

Keyword(s):

Boolean Algebras ◽

Equational Class ◽

Combinatorial Properties ◽

Varieties Of Lattices ◽

Nullary Operation ◽

New Construction ◽

Fixed Type

A new construction of algebras called a mapping extension of an algebra is here introduced. The construction yields a generalization of some classical constructions such as the nilpotent extension of an algebra, inflation of a semigroup but also the square extension construction introduced recently for idempotent groupoids. The mapping extension construction is defined for algebras of any fixed type, however nullary operation symbols are here not admitted. It is based on the notion of a retraction and some system of mappings. A mapping extension of a given algebra is constructed as a counterimage algebra by a specially defined retraction. Varieties of algebras satisfying an identity φ(x) ≈ x for a term φ not being a variable (such as varieties of lattices, Boolean algebras, groups and rings) are especially interesting because for such a variety [Formula: see text], all mapping extensions by φ of algebras from [Formula: see text] form an equational class. In the last section, combinatorial properties of the mapping extension construction are considered.

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