Infinite antichains in semilattices

Abstract

In this paper we consider infinite antichains and the semilattices that they generate, mainly in the context of continuous semilattices. Conditions are first considered that lead to the antichain generating a copy of a countable product of the two-element semilattice. Then a special semilattice, called Δ, is defined, its basic properties developed, and it is shown in our main result that a continuous semilattice with an infinite antichain converging to a larger element contains a semilattice copy of Δ. The paper closes with a consideration of countable antichains that converge to a lower element or a parallel element and the kinds of semilattices generated in this context.