A metrizable Lawson semitopological semilattice with non-closed partial order

TL;DR

This paper constructs a metrizable Lawson semitopological semilattice where the partial order is not closed, answering an open problem in the field.

Contribution

It provides the first example of a metrizable Lawson semitopological semilattice with a non-closed partial order, resolving a previously posed question.

Findings

Constructed a specific example of such a semilattice.

Showed the partial order in this example is not closed.

Resolved an open problem in the theory of semitopological semilattices.

Abstract

We construct a metrizable Lawson semitopological semilattice $X$ whose partial order $\le_X=\{(x,y)\in X\times X:xy=x\}$ is not closed in $X\times X$. This resolves a problem posed earlier by the authors.