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On ideals and filters in posets. (English) Zbl 1389.06006
Summary: The subject-matter of this paper is ideal theory in quite arbitrary posets. We offer an overview of the state of the art and several new results. We begin with the ideal-like theory of order ideals devised by P. V. Venkatanarasimhan [Math. Ann. 185, 338–348 (1970; Zbl 0182.33902); Proc. Am. Math. Soc. 28, 9–17 (1971; Zbl 0218.06002)]. We have identified four generalizations of (semi)lattice ideals to arbitrary posets: Birkhoff’s normal ideals, Frink ideals, the ideals suggested by Venkatanaasimhan (V-ideals), and those used by R. Balbes and P. Dwinger [Distributive lattices. Columbia: University of Missouri Press (1975)]; several common features of them are pointed out in Section 2. Then we present an unpublished work of T. Katriňák [“Distributive teilweise geordnete Mengen”, Lecture notes], including a characterization of those posets whose lattice of V-ideals is distributive. In Section 4 we prove several properties of Balbes-Dwinger ideals. The main results are an intrinsic description of the ideal generated by a set and the fact that in the class of distributive posets (more general than semilattices) every maximal ideal is prime and the ideal lattice is distributive. The conclusions include suggestions for further surveys.

06A11 Algebraic aspects of posets
06A12 Semilattices