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Congruences on semilattices with section antitone involutions. (English) Zbl 1243.06004

A semilattice with section antitone involutions is an algebra \(S=(S; \vee, 1, (^a)_{a\in S})\) such that \((S;\vee, 1)\) is a join-semilattice with 1 and to each \(a\in S\) there exists an antitone involution \(x\rightarrow x^a\) on the interval (= section) \([a,1]\), i.e. \((x^a)^a=x\) and \(a\leq x\leq y\) implies \(y^a\leq x^a.\)
The author investigates congruences on semilattices with section antitone involutions (= semilattices with SAI). Here are some results: (1) Let \(S\) be a semilattice with SAI. Then every congruence relation \(\Theta\) on \(S\) is uniquely determined by its kernel \([1]\Theta.\) (2) Every semilattice \(S\) with SAI is congruence-distributive. (3) Let \(S\) be a semilattice with SAI having a smallest element 0. Then \(S\) is congruence-permutable.

MSC:

06A12 Semilattices
06D35 MV-algebras
08A30 Subalgebras, congruence relations
08B10 Congruence modularity, congruence distributivity
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