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The fundamental theorem of finite semidistributive lattices. (English) Zbl 1485.06003

It is known that a finite poset \(L\) is a distributive lattice if and only if it is isomorphic to \(Downset(P)\) for some finite poset \(P\). The authors prove a similar result for semidistributive lattices. Theorem 2.1. A finite poset \(L\) is a semidistributive lattice if and only if there exists a set \(S\) with some additional structure, such that \(L\) is isomorphic to the admissible subsets of \(S\) ordered by inclusion. In this case, \(S\) and its additional structure are uniquely determined by \(L\).
All of these concepts are defined in the paper. The authors also study an infinite case and prove a number of theorems on semidistributive lattices and the mentioned construction.

MSC:

06B05 Structure theory of lattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B15 Representation theory of lattices
06D75 Other generalizations of distributive lattices
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