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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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