Meet-irreducible elements in implicative lattices

Smith, Dorothy P.

doi:10.1090/S0002-9939-1972-0291035-3

Meet-irreducible elements in implicative lattices

Author: Dorothy P. Smith
Journal: Proc. Amer. Math. Soc. 34 (1972), 57-62
MSC: Primary 06A35
DOI: https://doi.org/10.1090/S0002-9939-1972-0291035-3
MathSciNet review: 0291035
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A characterization of meet-irreducible elements and atoms in an implicative lattice is obtained and used to derive the following theorems. A complete lattice is implicative and every element has a meet-irreducible decomposition if and only if there are enough principal prime relative annihilator ideals to separate distinct elements. The MacNeille completion of an implicative lattice is an implicative lattice; furthermore the embedding preserves relative pseudocomplements, meet-irreducible elements and atoms.

V. K. Balachandran, On complete lattices and a problem of Birkhoff and Frink, Proc. Amer. Math. Soc. 9 (1955), 548–553. MR 72852, DOI https://doi.org/10.1090/S0002-9939-1955-0072852-1
Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
Peter Crawley, Regular embeddings which preserve lattice structure, Proc. Amer. MAth. Soc. 13 (1962), 748–752. MR 0140451, DOI https://doi.org/10.1090/S0002-9939-1962-0140451-2
Nenosuke Funayama, On the completion by cuts of distributive lattices, Proc. Imp. Acad. Tokyo 20 (1944), 1–2. MR 14063

V. Glivenko, Sur quelques points de la logique de Brouwer, Bull. Acad. Sci. Belgique 15 (1929), 183-188.

H. M. MacNeille, Partially ordered sets, Trans. Amer. Math. Soc. 42 (1937), no. 3, 416–460. MR 1501929, DOI https://doi.org/10.1090/S0002-9947-1937-1501929-X
Mark Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970), 377–386. MR 256951
J. C. C. McKinsey and Alfred Tarski, On closed elements in closure algebras, Ann. of Math. (2) 47 (1946), 122–162. MR 15037, DOI https://doi.org/10.2307/1969038
William C. Nemitz, Implicative semi-lattices, Trans. Amer. Math. Soc. 117 (1965), 128–142. MR 176944, DOI https://doi.org/10.1090/S0002-9947-1965-0176944-9

T. Newman, Homomorphic images of compactly generated lattices, Doctoral Dissertation, University of Texas, Austin, Tex., 1967 (unpublished).

Helena Rasiowa and Roman Sikorski, The mathematics of metamathematics, Monografie Matematyczne, Tom 41, Państwowe Wydawnictwo Naukowe, Warsaw, 1963. MR 0163850

M. H. Stone, Topological representation of distributive lattices and Brouwerian logics, Casopis Mat. Fys. 67 (1937), 1-25.

Morgan Ward, Structure residuation, Ann. of Math. (2) 39 (1938), no. 3, 558–568. MR 1503424, DOI https://doi.org/10.2307/1968634