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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
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