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Post almost distributive lattices. (English) Zbl 1438.06028

Summary: The concepts of chain based almost distributive lattices (ADLs) were introduced in our earlier papers. In this paper, we introduce a Post almost distributive lattice (Post ADL) with a chain base and we prove the basic properties and facts on a Post ADL related to a \({P_0}\)-ADL, a \({P_1}\)-ADL, a \({P_2}\)-ADL and a \(B\)-ADL (\(BL\)-ADL). We prove a necessary and sufficient condition for an ADL to become a Post ADL and finally prove that a \({P_2}\)-ADL of order \(n\) is almost isomorphic to a direct product of Post ADLs of maximum order \(n\).

MSC:

06D25 Post algebras (lattice-theoretic aspects)
03G20 Logical aspects of Łukasiewicz and Post algebras
06D75 Other generalizations of distributive lattices
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