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Remarks on semiseparation of lattices. (English) Zbl 0733.28006
Let X be a non-empty set. For a lattice we refer to any lattice of subsets of X containing $$\emptyset$$ and X. If $${\mathcal L}_ 1$$ and $${\mathcal L}_ 2$$ are two lattices such that $${\mathcal L}_ 1\subseteq {\mathcal L}_ 2,$$ then $${\mathcal L}_ 1$$ semiseparates $${\mathcal L}_ 2$$ if $$L_ 1\in {\mathcal L}_ 1,$$ $$L_ 2\in {\mathcal L}_ 2$$ and $$L_ 1\cap L_ 2=\emptyset$$ imply that there exists $$L\in {\mathcal L}_ 1$$ such that $$L_ 2\subseteq L$$ and $$L_ 1\cap L=\emptyset.$$ Let $${\mathcal L}$$ be a lattice. Then (1) $${\mathcal L}$$ is a delta lattice if $${\mathcal L}$$ is closed under countable intersections; (2) $${\mathcal L}$$ is disjunctive if $$x\in X,$$ $$L_ 1\in {\mathcal L}$$ and $$x\not\in L_ 1$$ imply that there exists $$L_ 1\in {\mathcal L}$$ such that $$x\in L_ 2$$ and $$L_ 1\cap L_ 2=\emptyset;$$ (3) The symbol I($${\mathcal L})$$ denotes the set of all non-trivial $$\{0,1\}-$$valued finitely additive set functions defined on the algebra A($${\mathcal L})$$ generated by $${\mathcal L}$$; (4) An element $$\mu \in I({\mathcal L})$$ is $${\mathcal L}$$-regular if $$\mu (A)=\sup \{\mu (L):\;L\in {\mathcal L}\text{ and } L\subseteq A\}$$ for all $$A\in A({\mathcal L});$$ (5) The symbol $$I_ R({\mathcal L})$$ denotes the set of all $${\mathcal L}$$-regular elements of I($${\mathcal L})$$; (6) The symbol $$I^{\sigma}_ R({\mathcal L})$$ denotes the set of all countably additive elements of $$I_ R({\mathcal L})$$; (7) An $${\mathcal L}$$- filter is a subset F of $${\mathcal L}$$ satisfying the following conditions: $$i)\quad \emptyset \not\in {\mathcal F},$$ ii) $${\mathcal F}$$ is closed under finite intersections, and iii) If $$F\in {\mathcal F},$$ $$L\in {\mathcal L}$$ and $$F\subseteq L,$$ then $$L\in {\mathcal F};$$ (8) $${\mathcal L}$$ is an I-lattice if every $${\mathcal L}$$-filter with the countable intersection property is contained in an $${\mathcal L}$$-ultrafilter with the countable intersection property. It is easy to verify that the set $$\{\{\mu \in I_ R({\mathcal L}):\;\mu (L)=1:\;L\in {\mathcal L}\}$$ is a base for the closed sets of a topology on $$I_ R({\mathcal L})$$, called the Wallman topology on $$I_ R({\mathcal L}).$$
Now the main results of the paper under review can be stated as follows:
Theorem A. Let $${\mathcal L}_ 1$$ and $${\mathcal L}_ 2$$ be two lattices such that $${\mathcal L}_ 1\subseteq {\mathcal L}_ 2$$, let $$\mu \in I_ R({\mathcal L}_ 1)$$ and, for every $$E\subseteq X,$$ let $$\mu '(E)=\inf \{\mu (X\setminus L_ 1):\;L_ 1\in {\mathcal L}_ 1\text{ and } E\subseteq X\setminus L_ 1\}$$ and $${\tilde \mu}(E)=\inf \{\mu (L_ 1):\;L_ 1\in {\mathcal L}_ 1\text{ and } E\subseteq L_ 1\}.$$ Then $$\mu '| {\mathcal L}_ 2={\tilde \mu}| {\mathcal L}_ 2$$ if and only if $${\mathcal L}_ 1$$ semiseparates $${\mathcal L}_ 2.$$
Theorem B. Let $${\mathcal L}_ 1$$ and $${\mathcal L}_ 2$$ be two lattices such that $${\mathcal L}_ 1\subseteq {\mathcal L}_ 2$$, $${\mathcal L}_ 1$$ is delta and an I-lattice, $${\mathcal L}_ 2$$ is disjunctive and, for every $$\mu \in I_ R({\mathcal L}_ 2),$$ the restriction $$\mu | A({\mathcal L}_ 1)$$ is $${\mathcal L}_ 1$$-regular. If the map $$\chi (\mu)=\mu | A({\mathcal L}_ 1)$$ is closed with respect to the corresponding Wallman topologies, then $${\mathcal L}_ 1$$ semiseparates $${\mathcal L}_ 2$$.
##### MSC:
 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 28A60 Measures on Boolean rings, measure algebras
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