## The fundamental theorem of finite semidistributive lattices.(English)Zbl 07383336

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 proved 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 study also an infinite case and prove a number of theorems on semidistributive lattices and the mentioned construction.

### MSC:

 08B05 Equational logic, Mal’tsev conditions 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06B15 Representation theory of lattices 06D75 Other generalizations of distributive lattices

### Keywords:

distributive lattice; semidistributive lattice
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### References:

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