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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 $$L$$.
##### MSC:
 06B10 Lattice ideals, congruence relations
##### Keywords:
lattice; weak direct factor
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##### References:
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