Pseudo-complements in posets

Venkatanarasimhan, P. V.

doi:10.1090/S0002-9939-1971-0272687-X

Pseudo-complements in posets
HTML articles powered by AMS MathViewer

by P. V. Venkatanarasimhan PDF

Proc. Amer. Math. Soc. 28 (1971), 9-17 Request permission

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.

References

V. K. Balachandran, A characterization for complete Boolean algebras, J. Madras Univ. B 24 (1954), 273–278. MR 65525
Garrett Birkhoff, Lattice Theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. 25, American Mathematical Society, New York, N. Y., 1948. MR 0029876
Orrin Frink Jr., Representations of Boolean algebras, Bull. Amer. Math. Soc. 47 (1941), 755–756. MR 5095, DOI 10.1090/S0002-9904-1941-07554-3
Orrin Frink, Ideals in partially ordered sets, Amer. Math. Monthly 61 (1954), 223–234. MR 61575, DOI 10.2307/2306387
Orrin Frink, Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962), 505–514. MR 140449
G. Szász, On complemented lattices, Acta Sci. Math. (Szeged) 19 (1958), 77–81. MR 95792
Jules Varlet, Contribution à l’étude des treillis pseudo-complémentés et des treillis de Stone, Mém. Soc. Roy. Sci. Liège Coll. in-8$^\circ$ (5) 8 (1963), no. 4, 71 (French). MR 164909