On separation of lattices (Q810175)

Let X be a non-empty set. A lattice refer to any lattice of subsets of X containing \(\emptyset\) and X. Let \({\mathcal L}\) be a lattice. Then: (1) \({\mathcal L}\) is disjunctive if \(x\in X,\) \(L_ 1\in {\mathcal L}\) and \(x\not\in L_ 1\) imply that there exists \(L_ 2\in {\mathcal L}\) such that \(x\in L_ 2\) and \(L_ 1\cap L_ 2=\emptyset;\) (2) \({\mathcal L}\) is normal if \(L_ 1,L_ 2\in {\mathcal L}\) and \(L_ 1\cap L_ 2=\emptyset\) imply that there exist \(L_ 3,L_ 4\in {\mathcal L}\) such that \(L_ 1\subseteq X\setminus L_ 3,\) \(L_ 2\subseteq X\setminus L_ 4\) and \((X\setminus L_ 3)\cap (X\setminus L_ 4)=\emptyset;\) (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 called \({\mathcal L}\)-regular if \(\mu (A)=\sup \{\mu (L):\;L\in {\mathcal L}\text{ and } L\subseteq A\}\) for all \(A\in A({\mathcal L});\) and (5) The symbol \(I_ R({\mathcal L})\) denotes the set of all \({\mathcal L}\)-regular members of I(\({\mathcal L})\). Let \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) be two lattices. We say that: (a) \({\mathcal L}_ 1\) separates \({\mathcal L}_ 2\) if \(L_ 0,L_ 2\in {\mathcal L}_ 2\) and \(L_ 0\cap L_ 2=\emptyset\) imply that there exist \(L_ 1,L_ 3\in {\mathcal L}_ 1\) such that \(L_ 0\subseteq L_ 1,\) \(L_ 2\subseteq L_ 3\) and \(L_ 1\cap L_ 3=\emptyset;\) (b) \({\mathcal L}_ 1\) 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 \(L\cap L_ 1=\emptyset.\) 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 \({\mathcal L}_ 1\subseteq {\mathcal L}_ 2\) and \({\mathcal L}_ 1\) separates \({\mathcal L}_ 2\). Then \({\mathcal L}_ 1\) is normal if and only if \({\mathcal L}_ 2\) is normal. Theorem B. Let \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) be two lattices such that \({\mathcal L}_ 1\) is normal, \({\mathcal L}_ 2\) is disjunctive and \({\mathcal L}_ 1\subseteq {\mathcal L}_ 2.\) If, for every \(\mu \in I_ R({\mathcal L}_ 2)\) the restriction \(\mu | A({\mathcal L}_ 1)\) is \({\mathcal L}_ 1\)-regular, then \({\mathcal L}_ 1\) semiseparates \({\mathcal L}_ 2\).