Tolerances, covering systems, and the axiom of choice
George Grätzer, Wenzel, G. H. · Czech digital mathematics library · 1989
A tolerance relation of an algebra is a binary relation that is reflexive, symmetric, and has the Substitution Property.A number of authors (I.Chajda, G. Cz6dli, L. Klukovits, J. Niederle, I. Rosenberg, D. Schweigert, B. Zelinka, and the present authors) investigated how tolerances can be described by the system of blocks (maximal connected subsets).In this paper we show how to modify known results from idempotent algebras to arbitrary algebras.We prove the known characterization for lattices without the Axiom of Choice.For lattices with the Chain Condition, G. Cz&lli and L. Klukovits obtained a much better result.We generalize their result to arbitrary lattices, again avoiding the use of the Axiom of Choice.Finally, we show that for semilattices, the existence of a tolerance-block is equivalent to the Axiom of Choice.