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A completion for distributive nearlattices. (English) Zbl 1455.06004

By a polarity a triple \((X,Y,R)\) is meant where \(X,Y\) are nonempty sets and \(R\) is a binary relation between \(X\) and \(Y\). For a poset \(P\), a completion of \(P\) is a pair \((L,e)\) where \(L\) is a complete lattice and \(e\) is an order embedding of \(P\) into \(L\). A collection \(F\) of upsets of \(P\) is \(standard\) if it contains all principal filters of \(P\), dually for a collection \(I\) of downsets of \(P\). A polarity is \(standard\) if it is of the form \((F,I,R)\) for standard collections \(F\) and \(I\). The authors introduce the so-called \((F,I)\)-compact and \((F,I)\)-dense polarities and the so-called \((F,I)\)-completion They prove that every distributive nearlattice can be embedded into a complete distributive lattice via an \((F,I)\)-completion and presented a connection with free distributive lattice extension. They study how an \(n\)-ary operation can be extended on a distributive nearlattice.

MSC:

06A12 Semilattices
06B23 Complete lattices, completions
03G10 Logical aspects of lattices and related structures
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06D05 Structure and representation theory of distributive lattices
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