Order-generation in posets and convolution of closure operators

Abstract

Motivated by the Hofmann--Lawson theorem, which states that every continuous lattice is inf-generated by its irreducible elements, we explore how to represent posets by extreme points with respect to a closure operator. For this purpose, we introduce the convolution product of closure operators, and prove that the Krein--Milman property can be transferred from one collection of subsets to another by convolution. This result underpins two new representation theorems of topological flavor, which generalize existing ones, even in posets lacking lattice or semilattice structures. We also prove a third representation theorem: given a poset equipped with a closure operator $\mathfrak{c}$ with adequate properties, we show that the set of kit points, defined as an extension of compact points, has the Krein--Milman property with respect to the convolution product of $\mathfrak{c}$ with the dual Alexandrov operator $\uparrow\!\! \cdot$; moreover, every kit point is sup-generated by a unique antichain of compact points, finite if $\mathfrak{c}$ is finitary.