zbMATH — the first resource for mathematics

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)

MSC:
 06B05 Structure theory of lattices 06B15 Representation theory of lattices
Full Text:
References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.