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Tolerances and congruences on lattices. (English) Zbl 0598.06004
Let \({\mathcal L}\) be a bounded lattice, let \({\mathcal S}\) be a semigroup of isotone mappings on \({\mathcal L}\). Let R be a reflexive and symmetric binary relation on \({\mathcal L}\). If aRb\(\Rightarrow \phi (a)R\phi (b)\) for all \(\phi\in {\mathcal S}\), then R is called a set-theoretic \({\mathcal S}\)- tolerance. If moreover aRb, cRd imply (a\(\vee c)R(b\vee d)\), (a\(\wedge c)R(b\wedge d)\), then R is an \({\mathcal S}\)-tolerance; if moreover R is transitive, then R is an \({\mathcal S}\)-congruence. All set-theoretic \({\mathcal S}\)-tolerances (or \({\mathcal S}\)-tolerances, or \({\mathcal S}\)-congruences) on L form a complete lattice ST(\({\mathcal S},{\mathcal L})\) (or LT(\({\mathcal S},{\mathcal L})\), or Con(\({\mathcal S},{\mathcal L})\) respectively).
The paper studies the lattice properties of these lattices. A particular attention is paid to Stone lattices. The concepts of a complementary tolerance and of a complete tolerance are introduced and studied.
Reviewer: B.Zelinka

MSC:
06B10 Lattice ideals, congruence relations
08A30 Subalgebras, congruence relations
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