Lattices of compatible relations

Archivum Mathematicum

Ivan Chajda Lattices of compatible relations Archivum Mathematicum, Vol. 13 (1977), No. 2, 89--95

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ARCH. MATH. 2, SCRIPTA FAC SCI. NAT. UJEP BRUNENSIS XIII: 89—96, 1977

LATTICES OF COMPATIBLE RELATIONS IVAN CHAJDA, Pferov (Received October 4, 1976)

As it is shown in [3], [4], [5] and [6], various results on congruences on algebras can be generalized also for other types of relations. The aim of this paper is to show some of common properties of lattices of relations and give mutual interrelations among these lattices. By 81 = denote an algebra with a base set A and the set of fundamental operations F. A binary relation R on A (i.e. R e A xA) is called to be compatible on A9 if for each n-ary fe F and arbitrary ai9 bte A (i = 1,..., n) the following implication is true: eR

for / = 1, ...,«=> {f(al9...,

an)9f(bi9...,

bn)} e R.

By Comp(8l) or ^(81) or &>(%) or ^(81) or J2\T(8l) or (Stl)', H s R}. For H = {} abbreviate Rp{{}) by R&(a, b) and call it the principal gP-relation generated by a, b Remark. Evidently, R^(H) e ^(SS) for every 0 # Hg AxA and the principal ^-relation is a generalization of a principal congruence (see [1]) and a minimal tolerance (see [4]). Theorem 4. Let 2t = be an algebra, a, be A and 0> e A0. Then = Comp, (b) flf = 6* 0r a{ = t4(fl), bf =- tt(b)for 0> ^ M,

(c) {flf, bt} = {t/4 *ivb)}f0r ^ = se, (d) fl, = bi or {ai9 bi} = {ti(a), ti(b)}for 0> = if,T. (2) there exist a0, ...,ane A and unary algebraic functions (see [1])