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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Pseudo-complements in posets


Author: P. V. Venkatanarasimhan
Journal: Proc. Amer. Math. Soc. 28 (1971), 9-17
MSC: Primary 06.35
MathSciNet review: 0272687
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Abstract: In this paper a theory of pseudo-complements is developed for posets (partially ordered sets). The concepts of ideal and semi-ideal are introduced for posets and a few results about them are obtained. These results together with known results about pseudo-complements in distributive lattices lead to the main results. It is proved that if in a pseudo-complemented semilattice or dual semilattice every element is normal, then it is a Boolean algebra. Using this result new proofs for two known theorems are obtained. The existence of maximal ideals in posets is established and it is shown that the dual ideal of dense elements of a poset with 0 is the product of all the maximal dual ideals.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0272687-X
PII: S 0002-9939(1971)0272687-X
Keywords: Poset, lattice, semilattice, Boolean algebra, ideal, semiideal, complement, relative-complement, pseudo-complement, quasi-complement, normal element, dense element
Article copyright: © Copyright 1971 American Mathematical Society