For a Hausdorff topologized semilattice X its Lawson number \(\bar{\Lambda }(X)\) is the smallest cardinal \(\kappa \) such that for any distinct points \(x,y\in X\) 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 \(x\in X\) 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 \(C\subseteq X\) has \(\inf C\in {\overline{C}}\) and \(\sup C\in {\overline{C}}\). We prove that for any complete subsemilattice X of an \(\omega \)-Lawson semitopological semilattice Y, the partial order \(\le _X=\{(x,y)\in X\times X:xy=x\}\) of X is closed in \(Y\times Y\) and hence X is closed in Y. This implies that for any continuous homomorphism \(h:X\rightarrow Y\) from a complete topologized semilattice X to an \(\omega \)-Lawson semitopological semilattice Y the image h(X) is closed in Y.

对于Hausdorff拓扑半格X，其劳森数 \（\ bar {\ Lambda}（X）\）是最小基数\（\ kappa \），因此对于任何不同点\（x，y \ X \）都存在家庭\（\ mathcal U \）的封闭社区的X中X，使得\（| \ mathcal U | \文件\卡帕\）和\（\ bigcap \ mathcal U \）中的subsemilattice X不包含ÿ。因此，\（\ bar {\ Lambda}（X）\ le \ bar {\ psi}（X）\），其中\（\ bar {\ psi}（X）\）是最小基数\（\ kappa \）这样，对于任何点\（X \在X \）存在一个家庭\（\ mathcal U \）的封闭社区的X中X，使得\（| \ mathcal U | \文件\卡帕\）和\（\ bigcap \ mathcal U = \ {x \} \）。我们证明了当且仅当\（\ bar {\ Lambda}（X）= 1 \）时，紧Hausdorff半拓扑半格X才是Lawson（即，具有由半子半词组成的拓扑的基础）。每个Hausdorff拓扑半晶格X的Lawson数为\（\ bar {\ Lambda}（X）\ le \ omega \）。另一方面，对于任何无限基数\（\ lambda \）我们构造一个Hausdorff零维半拓扑半格X，使得\（| X | = \ lambda \）和\（\ bar {\ Lambda}（X）= \ bar {\ psi}（X）= \ mathrm {cf} （\ lambda）\）。被拓扑化的半晶格X称为（i）\（\ omega \） - Lawson if \（\ bar {\ Lambda}（X）\ le \ omega \） ; （ⅱ）完全如果每个非空链\（C \ subseteq X \）具有\（\ INFÇ\在{\划线{C}} \）和\（\ SUPÇ\在{\划线{C}} \）。我们证明了对任何完整subsemilattice X的的\（\欧米茄\） -Lawson semitopological半格ÿ中，偏序\（\文件_X = \ {（X，Y）\在X \倍X：XY = X \} \）的X中被关闭\（Y \倍Y \） ，因此X在关闭ÿ。这意味着对于任何连续同态\（H：X \ RIGHTARROW Y \）从一个完整的拓扑结构化半格X到\（\欧米加\） -Lawson semitopological半格ý图像ħ（X）中被关闭ÿ。