Chajda, Ivan Congruences on semilattices with section antitone involutions. (English) Zbl 1243.06004 Discuss. Math., Gen. Algebra Appl. 30, No. 2, 207-215 (2010). 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. Reviewer: T. Katriňák (Bratislava) Cited in 1 Document MSC: 06A12 Semilattices 06D35 MV-algebras 08A30 Subalgebras, congruence relations 08B10 Congruence modularity, congruence distributivity Keywords:join-semilattice; section; antitone involution; congruence kernel; congruence distributivity; 3-permutability PDFBibTeX XMLCite \textit{I. Chajda}, Discuss. Math., Gen. Algebra Appl. 30, No. 2, 207--215 (2010; Zbl 1243.06004) Full Text: DOI Link