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0-distributive posets. (English) Zbl 1289.06002

Generalizing the notion of 0-distributive lattice and following the definition introduced by V. V. Joshi and B. N. Waphare [Math. Bohem. 130, No. 1, 73–80 (2005; Zbl 1112.06001)], a poset \(P\) with minimal element 0 is called 0-distributive if, for all elements \(x\), \(y\) and \(z\) in \(P\) the two conditions \(\{x,y\}^l=\{0\}\) and \(\{x,z\}^l=\{0\}\) imply that \(\{x,\{y,z\}^u\}^l=\{0\}\), where, for every subset \(A\) of \(P\), \(A^l=\{x\in P \mid x \leq a \text{ for every } a \in A\}\) is the lower cone of \(A\) and \(A^u\) is the dual notion. In this paper the authors give various characterizations of 0-distributive posets, by means of ideals and semiprime ideals. They also introduce the notion of semi-atom in a poset (which is much more general than the notion of atom) and the notion of semi-atomic poset. They get several characterizations of both concepts.

MSC:

06A06 Partial orders, general
06D75 Other generalizations of distributive lattices

Citations:

Zbl 1112.06001