The Lawson number of a semitopological semilattice

Abstract

For a Hausdorff topologized semilattice X its Lawson numberbar{Lambda }(X) is the smallest cardinal kappa such that for any distinct points x,yin X there exists a family mathcal U of closed neighborhoods of x in X such that |mathcal U|le kappa and bigcap mathcal U is a subsemilattice of X that does not contain y. It follows that bar{Lambda }(X)le bar{psi }(X), where bar{psi }(X) is the smallest cardinal kappa such that for any point xin X there exists a family mathcal U of closed neighborhoods of x in X such that |mathcal U|le kappa and bigcap mathcal U={x}. We prove that a compact Hausdorff semitopological semilattice X is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if bar{Lambda }(X)=1. Each Hausdorff topological semilattice X has Lawson number bar{Lambda }(X)le omega . On the other hand, for any infinite cardinal lambda we construct a Hausdorff zero-dimensional semitopological semilattice X such that |X|=lambda and bar{Lambda }(X)=bar{psi }(X)=mathrm {cf}(lambda ). A topologized semilattice X is called (i) omega -Lawson if bar{Lambda }(X)le omega ; (ii) complete if each non-empty chain Csubseteq X has inf Cin {overline{C}} and sup Cin {overline{C}}. We prove that for any complete subsemilattice X of an omega -Lawson semitopological semilattice Y, the partial order le _X={(x,y)in Xtimes X:xy=x} of X is closed in Ytimes Y and hence X is closed in Y. This implies that for any continuous homomorphism h:Xrightarrow Y from a complete topologized semilattice X to an omega -Lawson semitopological semilattice Y the image h(X) is closed in Y.