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Minimum implicational basis for \(\wedge\)-semidistributive lattices. (English) Zbl 1185.06009

Summary: For a \(\wedge\)-semidistributive lattice \(L\), we study some particular implicational systems and show that the cardinality of a minimum implicational basis is polynomial in the size of join-irreducible elements of the lattice \(L\). We also provide a polynomial-time algorithm to compute a minimum implicational basis for \(L\).

MSC:

06D75 Other generalizations of distributive lattices
03G10 Logical aspects of lattices and related structures
68Q25 Analysis of algorithms and problem complexity
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