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Direct decompositions of atomistic algebraic lattices. (English) Zbl 0818.06004
A lattice is atomistic if each of its elements is a join of atoms. An element \(x\) of a complete lattice \(L\) is strictly join-irreducible if \(x= \bigvee X\) implies \(x\in X\) for any subset \(X\subseteq L\). A lattice \(L\) is called a \(V_ 1\)-lattice if each its elements is the join of strictly join-irreducible elements of \(L\). Main results: Theorem 1. Every atomistic algebraic lattice is a direct product of directly indecomposable (atomistic algebraic) lattices. Theorem 2. Every algebraic \(V_ 1\)-lattice is a direct product of directly indecomposable (algebraic) \(V_ 1\)-lattices.
Reviewer: I.Chajda (Přerov)

06B05 Structure theory of lattices
06B15 Representation theory of lattices
Full Text: DOI
[1] Bennet, M. K.,Separation conditions on convexity lattices, Springer Lecture Notes in Mathematics1149 (1987), 22-37. · doi:10.1007/BFb0098453
[2] Bennett, M. K. andBirkhoff, G.,Convexity lattices, Algebra Universalis20 (1985), 1-26. · Zbl 0566.06005 · doi:10.1007/BF01236802
[3] Birkhoff, G.,Lattice Theory, 3rd ed., AMS, Providence, RI, 1967.
[4] Filippov, N. D.,Projectivity of lattices, Matem. Sb.70 (1966), 36-54.
[5] Grätzer, G.,General Lattice Theory. Birkhäuser Verlag, Basel, 1978. · Zbl 0385.06015
[6] Libkin, L.,Direct product decompositions of lattices, closures and relation schemes, Discrete Mathematics 112 (1993), 119-138. · Zbl 0780.06003 · doi:10.1016/0012-365X(93)90228-L
[7] Libkin, L. andMuchnik, I.,The lattices of subsemilattices of a semilattice, Algebra Universalis31 (1993), 252-255. · Zbl 0797.06003 · doi:10.1007/BF01236520
[8] Richter, G.,On the structure of lattices in which every element is a join of join-irreducible elements. Periodica Mathematica Hungarica13 (1982), 47-69. · Zbl 0484.06008 · doi:10.1007/BF01848096
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