Complemented tolerances on lattices. (English) Zbl 0536.06007

Chajda, Ivan; Zelinka, Bohdan

Čas. Pěst. Mat. 109, 54-59 (1984).

A tolerance relation on a lattice L is a reflexive symmetric binary relation T having the property that aTb, cTd together imply that \((a\vee c)T(b\vee d)\) and \((a\wedge c)T(b\wedge d).\) If the tolerance relations on L are partially ordered by set inclusion, they form a complete algebraic lattice called the tolerance lattice of L. The thrust of the paper is the proof of the assertion that if z,z’ are complementary elements of the bounded modular lattice L, and if T(0,z), T(0,z’) denote the minimal tolerance relations collapsing the quotients z/0, z’/0, then the following are equivalent: (a) z is central in L. (b) T(0,z) and T(0,z’) are complementary elements of the tolerance lattice of L.
Reviewer’s note: It is not difficult to show the result to be true for an arbitrary bounded lattice. The theorem is similar to a result of the reviewer on complemented congruences [Pac. J. Math. 73, 87-90 (1977; Zbl 0371.06005), p. 88].

Reviewer: M.F.Janowitz

MSC:

06B10	Lattice ideals, congruence relations
06C05	Modular lattices, Desarguesian lattices

Keywords:

tolerance lattice; bounded modular lattice; tolerance relations; complemented congruences

Citations:

Zbl 0371.06005