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On convex constructions in lattices. (English) Zbl 1083.06008
A pair \((L_1,L_2)\) is called an oriented convex decomposition of a lattice \((L,\leq)\) provided: \(L_1\) and \(L_2\) are proper sublattices of \(L\); \(L_1\cap L_2\neq\emptyset\); \(L_1\cup L_2 = L\); for any \(i,j\) with \(1\leq i\neq j\leq 2\), any \(v\in L_i\setminus L_j\) and any \(w\in L_j\setminus L_i\) such that \(v\leq w\) there exists \(x_0\in L_1\cap L_2\) satisfying \(v\leq x_0\leq w\); \(L_1\cap L_2\) is convex; \(L_1 =\{x\in L:\exists y\in L_1\cap L_2,\;x\leq y\}\); and \(L_2 = \{x\in L:\exists y\in L_1\cap L_2,\;y\leq x\}\). Suppose that the pair \((L_1,L_2)\) is an oriented convex decomposition of a lattice \((L,\leq)\). The author proves that if \(L_1\) and \(L_2\) are distributive (respectively, modular; respectively Brouwerian) lattices, then \(L\) is a distributive (respectively, modular; respectively Brouwerian) lattice. The author also obtains the following partial converse. If \(L\) is a Browerian lattice and \(L\) has a greatest element, then \(L_1\) and \(L_2\) are Browerian lattices. The author also investigates oriented convex decompositions for pseudo-complemented lattices.
MSC:
06B05 Structure theory of lattices
06C05 Modular lattices, Desarguesian lattices
06D05 Structure and representation theory of distributive lattices
06D15 Pseudocomplemented lattices
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