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On completion of section semicomplemented posets. (English) Zbl 1150.06001

Summary: In this paper, we prove that the completion by cuts of a section semicomplemented (in brief SSC) poset is section semicomplemented. It is shown that a finite distributive section semicomplemented poset need not be Boolean. However, its completion by cuts is a Boolean lattice. The concepts of normal ideals and subprojective ideals are introduced in posets and it is proved that the center of the completion by cuts of a section semicomplemented and dually section semicomplemented poset \(P\) is precisely the set of normal subprojective ideals of \(P\). Further, it is proved that in an SSC and SSC\(^*\) (the dual of SSC) poset \(P\), a principal ideal \(J_s\) is subprojective if and only if \(s\) is a neutral element of \(P\).

MSC:

06A06 Partial orders, general
06C15 Complemented lattices, orthocomplemented lattices and posets
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