Tolerances, covering systems, and the axiom of choice. (English) Zbl 0711.08002

Grätzer, G.; Wenzel, G. H.

Arch. Math., Brno 25, No. 1-2, 27-34 (1989).

By a block of the tolerance T on an algebra A is meant the maximal T- connected subset of A. A number of authors gave characterizations of blocks and of the system of all blocks of some tolerance on A. In this paper, the known characterization for lattices is proved without the Axiom of Choice. Some known results are modified from idempotent algebras to a general case. Finally, the authors show that for semilattices the existence of a tolerance block is equivalent to the Axiom of Choice.

Reviewer: I.Chajda

MSC:

08A30	Subalgebras, congruence relations
06B10	Lattice ideals, congruence relations
03E25	Axiom of choice and related propositions
08A05	Structure theory of algebraic structures

Keywords:

tolerance relation; covering system; tolerance block