The Lawson number of a semitopological semilattice

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