zbMATH — the first resource for mathematics

Weak direct factors of lattices. (English) Zbl 1164.06307
A sublattice \(A\) of a lattice \(L\) is defined to be a weak direct factor of \(L\) if there exists a lattice \(B\) with a least element 0 and an embedding \(\varphi \: L \rightarrow A\times B\) with \(\varphi (A)=A\times \{ 0\}\). An internal characterization of this notion is presented. In particular, weak direct factors of a distributive lattice are just its almost principal ideals. It is proved that if \(L\) is distributive or if it has a greatest element, then the system \(W(L)\) of all weak direct factor of \(L\) is a sublattice of the lattice of all ideals of \(L\).
06B10 Lattice ideals, congruence relations
Full Text: EuDML
[1] BIRKHOFF G.: Lattice Theory. (3rd, Amer. Math. Soc, Providence, RI, 1967. · Zbl 0153.02501
[2] GRÄTZER G.: General Lattice Theory. (2nd, Birkhauser Verlag, Basel-Boston-Berlin, 1998. · Zbl 0909.06002
[3] GRÄTZER G.-SCHMIDT E. T.: Ideals and congruence relations in lattices. Acta Math. Acad. Sci. Hungar. 9 (1958), 137-175. · Zbl 0085.02002 · doi:10.1007/BF02023870
[4] JAKUBÍK J.: Direct product decomposition of MV-algebras. Czechoslovak Math. J. 44 (119) (1994), 725-739. · Zbl 0821.06011 · eudml:31437
[5] JAKUBÍK J.: Atomicity of the Boolean algebra of direct factors of a directed set. Math. Bohem. 123 (1998), 145-161. · Zbl 0938.06011 · eudml:29458
[6] JAKUBÍK J.: Direct product decompositions of infinitely distributive lattices. Math. Bohem. 125 (2000), 341-354. · Zbl 0967.06004 · eudml:227203
[7] JAKUBÍK J.: On direct and subdirect product decompositions of partially ordered sets. Math. Slovaca 52 (2002), 377-395. · Zbl 1016.06002 · eudml:31827
[8] JAKUBÍK J.: Completely subdirect products of directed sets. Math. Bohem. 127 (2002), 71-81. · Zbl 0999.06002 · eudml:32618
[9] JAKUBÍK J.-CSONTOÓOVÁ M.: Convex isomorphisms of directed multilattices. Math. Bohem. 118 (1993), 359-378. · Zbl 0802.06008
[10] MAEDA F.: Kontinuierliche Geometrien. Springer Verlag, Berlin, 1958. · Zbl 0081.02601
[11] PLOŠČICA M.: Affine complete distributive lattices. Order 11 (1994), 385-390. · Zbl 0816.06010 · doi:10.1007/BF01108769
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.