Infinite distributive laws for the lateral operations on a Riesz space

Abstract

We prove analogues of the well known infinite distributive laws for the lateral infima and suprema instead of order ones. The proofs are more involved than that for the original laws. We show that one of the laws holds true whenever both sides of the equality are well defined. The other one is false in general, even if both sides are well defined, but true for finite sets. The proofs of two laws are completely different. The question of under what assumptions on the Riesz space and objects involved in, the second distributive law is valid for infinite sets, remains unsolved.