Staircases and a Congruence-Theoretical Characterization of Vector Spaces
We study so-called staircases in lattices and certain related lattice properties which are weaker than distributivity but stronger than meet- semidistributivity. Staircases are in some sense typical for finitely generated lattices of finite width which violate the ascending chain condition. A specific binary operation, defined in terms of staircases and involving (infinite) joins and. meets, turns out to be closely connected with the commutator in congruence lattices. Using this operation, we exhibit large meet-semidistributive varieties; furthermore, we characterize non-simple vector spaces, up to polynomial equivalence, by the property that they generate modular varieties and each nonzero congruence is an atom but not join-prime. Similar characterizations are given for infinite-dimensional vector spaces and, finally, for vector spaces over (commutative) fields.
KeywordsUniversal Algebra Congruence Lattice Modular Lattice Modular Variety Algebraic Lattice
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