On set-representable orthocomplemented difference lattices
An orthocomplemented difference lattice is an algebra \(\mathbf L=(L,\vee,\wedge,\Delta,{}',0,1)\) of type \((2,2,2,1,0,0)\) such that \((L,\vee,\wedge,{}',0,1)\) is an orthocomplemented lattice and the following identities are satisfied: \(\Delta\) is associative, \(x\Delta1\approx1\Delta x\approx x'\) and \(x\Delta y\le x\vee y\). \(\mathbf L\) is called set-representable if there exists a set \(X\) and a subset \(\Omega\) of \(2^X\) such that \(X\in\Omega\), \(A\Delta B\in\Omega\) for all \(A,B\in\Omega\) and \((L,\le,\Delta,0,1)\cong(\Omega,\subseteq,\Delta,\emptyset,X)\). It is proved that the class of set-representable orthocomplemented difference lattices is not locally finite. An orthomodular lattice is called concrete if it is isomorphic to a collection of subsets of a set with partial ordering given by set inclusion, orthocomplementation given by set complementation and finite orthogonal joins given by disjoint unions. An orthomodular lattice \(\mathbf M\) is said to be OML-embeddable into \(\mathbf L\) if \(\mathbf M\) can be embedded into the orthomodular lattice \((L,\vee,\wedge,{}',0,1)\). It is proved that not every concrete orthomodular lattice can be OML-embedded into an orthocomplemented difference lattice. Reviewer's remark: In the first reference ``Länger, H. M.'' should be ``Länger, H.''.
scientific article; zbMATH DE number 7259279
scientific article