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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.
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