Abstract
We generalize the notions of semidistributive elements and of prime ideals from lattices to arbitrary posets. Then we show that the Boolean prime ideal theorem is equivalent to the statement that if a posetP has a join-semidistributive top element then each proper ideal ofP is contained in a prime ideal, while the converse implication holds without any choice principle. Furthermore, the prime ideal theorem is shown to be equivalent to the following order-theoretical generalization of Alexander’s subbase lemma: If the top element of a posetP is join-semidistributive and compact in some subbase ofP then it is compact inP.
Similar content being viewed by others
References
Abian A.,A partial order generalization of Alexander’s subbase theorem. Rend. Circ. Mat. Palermo38 (1989), 271–276.
Alexander J.W.,Ordered sets, complexes, and the problem of bicompactification, Proc. Nat. Acad. Sci. U.S.A.22 (1939), 296–298.
Banaschewski B.,The power of the ultrafilter theorem. J. London Math. Soc. (2)27 (1983), 193–202.
Banaschewski B.,Prime elements from prime ideals. Order2 (1985), 211–213.
Erné M.,Distributivgesetze und die Dedekind’sche Schnittvervollständigung. Abh. Braunschweig. Wiss. Ges.33 (1982), 117–145.
Erné M.,Weak distributive laws and their role in lattices of congruences and equational theories. Alg. Univ.25 (1988), 290–321.
Erné M.,Distributors and Wallmanufacture. J. Pure and Appl. Algebra68 (1990), 109–125.
Frink O.,Ideals in partially ordered sets. Amer. Math. Monthly61 (1954), 223–234.
Gierz G., Hofmann K.H. Keimel K., Lawson J.D., Mislove M., Scott D.S.,A compendium of continuous lattices. Springer-Verlag, Berlin-Heidelberg-New York, 1980.
Gorbunov A.V., Tumanov V.I.,On the existence of prime ideals in semidistributive lattices. Alg. Univ.16 (1983), 250–252.
Jónsson B., Kiefer J.E.,Finite sublattices of a free lattice. Canad. J. Math.14 (1962), 487–497.
Klimovsky G.,Zorn’s theorem and the existence of maximal filters and ideals in distributive lattices. Rev. Un. Mat. Argentina18 (1958), 160–164.
Papert D.,Congruence relations in semilattices. J. London Math. Soc.39 (1964), 723–729.
Parovičenko I.I.,Topological equivalents of the Tihonov Theorem, Dokl. Akad. Nauk SSSR184 (1969), 38–39 = Soviet Math. Dokl.10 (1969), 33–34.
Rav Y.,Variants of Rado’s selection lemma and their applications. Math. Nachr.79 (1977), 145–155.
Rav Y.,Semiprime ideals in general lattices. J. Pure and Appl. Algebra56 (1989), 105–118.
Rubin H., Scott D.,Some topological theorems equivalent to the Boolean prime ideal theorem. Bull. Amer. Math. Soc.60 (1954), 389.
Scott D.,Prime ideal theorems for rings, lattices, and Boolean algebras. Bull. Amer. Math. Soc.60 (1954), 390.
Author information
Authors and Affiliations
Additional information
1980 Mathematics Subject Classification. Primary 06 A 10; secondary 06 D 05, 54 D 30.
Rights and permissions
About this article
Cite this article
Erné, M. Semidistributivity, prime ideals and the subbase lemma. Rend. Circ. Mat. Palermo 41, 241–250 (1992). https://doi.org/10.1007/BF02844668
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02844668