Quantifier theory on quasi-orthomodular lattices
Melvin F. Janowitz · Illinois Journal of Mathematics · 1965
IntroductionIn [7], we introduced the notion of a quantifier on an orthomodular lattice, and subject to mild assumptions of completeness, explicitly determined all quantifiers on an atomic orthomodular lattice [7, Theorem 7, p. 1245].The definition we gave of a quantifier can be extended to an arbitrary lattice L with 0 by agreeing that a mapping L --L shall be called a quantifier on L in case it satisfies" (Q1) 0=0.(Q2) e _ eq for all eeL.(Q3) (e h fq)q eq h f for all e, fei.Aside from the connection developed in [7, p. 1241], with P. Jordan's skew lattices, these mappings have a way of cropping up in a variety of situations.We present, herewith a few examples to illustrate this point.(i) In a lattice L with 0 and 1 there are always two quantifiers: the discrete quantifier--the identity map; the indiscrete quantifier defined by 0= 0, eq= lfore0.(ii) The column operator used by Halmos in his treatment of spectral multiplicity [6, p. 89] is a quantifier.(iii) In a Loomis dimension lattice the mapping a -a (see [12, p. 13]) is a quantifier.(iv) If L is a lattice with 0 and I having the property that for each a e L, a, ^{z e L z central, z >_ a} exists and is central, then , turns out to be a quantifier on L. We shall call , the central cover quantifier and L a central cover lattice.(v) Let L be a pseudo-complemented distributive lattice [1, pp.147- 148].If a* denotes the pseudo-complement of a, then a --> a** [1, Theorem 16, p. 148] is a quantifier.(vi) The closure operator of a topological space is a quantifier if and only if every open set is also closed (see [5, p. 43]).(vii) Let S be a commutative semigroup with 0, and let L denote the lattice of ideals of S. For an ideal I of S, define the radical of I by R(I) {x e S x e I for some positive integer n}.