拓扑半格的Lawson数
The Lawson number of a semitopological semilattice
摘要：
对于Hausdorff拓扑化半格$X$，其$Lawson\;\; number$ $\barΛ(X)$是最小基数$κ$，使得对于任何不同点$x,y\in X$，存在$X$中$x$的闭邻域的族$\mathcal U$，使得$|\mathcal U|\leκ$和$\bigcap\mathcal U$是不包含$y$的$X$的子半格。它遵循$\barΛ(X)\le\barψ(X)$，其中$\barψ(X)$是最小的基数$κ$，使得对于任何点$x\in X$，存在$X$中的$x$的封闭邻域的家族$\mathcal U$，使得$|\mathcal U|\leκ$和$\bigcap\mathcal U=\{x\}$。我们证明了一个紧Hausdorff拓扑半格$X$是Lawson（即，有一个由子半格组成的拓扑基）当且仅当$\barΛ(X)=1$。每个Hausdorff拓扑半格$X$都有Lawson数$\barΛ(X)\leω$ .另一方面，对任意无限基数$λ$，我们构造了一个Hausdorff零维拓扑半格$X$，使得$|X|=λ$和$\barΛ(X)=\barψ(X)=cf(λ)$ .一个拓扑化的半格$X$称为（i）$ω$-$Lawson$，如果$\barΛ(X)\leω$ ;（ii）$complete$，如果每个非空链$C\subset X$有$\inf C\in\overline{C}$和$\sup C\in\overline{C}$。证明了对于$ω$-Lawson拓扑半格$Y$的任何完备子半格$X$，$X$的偏序$\le_X=\{(x,y)\in X\times X:xy=x\}$在$Y\times Y$中闭，从而$X$在$Y$中闭.这意味着对于从完全拓扑半格$X$到$ω$-Lawson拓扑半格$Y$的任何连续同态$h:X\to Y$，像$h(X)$在$Y$中是闭的。
英文摘要：
For a Hausdorff topologized semilattice $X$ its $Lawson\;\; number$ $\barΛ(X)$ is the smallest cardinal $κ$ 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 $|\mathcal U|\leκ$ and $\bigcap\mathcal U$ is a subsemilattice of $X$ that does not contain $y$. It follows that $\barΛ(X)\le\barψ(X)$, where $\barψ(X)$ is the smallest cardinal $κ$ such that for any point $x\in X$ there exists a family $\mathcal U$ of closed neighborhoods of $x$ in $X$ such that $|\mathcal U|\leκ$ 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 $\barΛ(X)=1$. Each Hausdorff topological semilattice $X$ has Lawson number $\barΛ(X)\leω$. On the other hand, for any infinite cardinal $λ$ we construct a Hausdorff zero-dimensional semitopological semilattice $X$ such that $|X|=λ$ and $\barΛ(X)=\barψ(X)=cf(λ)$. A topologized semilattice $X$ is called (i) $ω$-$Lawson$ if $\barΛ(X)\leω$; (ii) $complete$ if each non-empty chain $C\subset X$ has $\inf C\in\overline{C}$ and $\sup C\in\overline{C}$. We prove that for any complete subsemilattice $X$ of an $ω$-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 $h:X\to Y$ from a compete topologized semilattice $X$ to an $ω$-Lawson semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$.
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