Distributivity in lattice ordered groups
Ján Jakubík · Czechoslovak Mathematical Journal · 1972
Higher degrees of distributivity in lattice ordered groups were studied by several authors [3], [4], [8], [9], [10], [14], [15].The resuhs of this paper are as follows.Let G be a lattice ordered group and let a, ß be any cardinals.It is proved that if G is a mixed product of lattice ordered groups A^ (i e/), then G is (a, jö)-distributive if and only if each Ai is (a, /?)-distributive.Let G be Archimedean and let E(G) be the Dedekind completion of G ; if G is (a, j5)-distributive and each non-trivial interval [a, b] of G contains a non-trivial interval [«i, bj with card [aj, bj ^ ß, then E(G) is (a, j5)-distributive.If G is (a, 2)-distributive and complete, then it is (a, a)-distributive (this is a partial solution of a problem by WEINBERG [14]).Assume that G is not completely distributive and denote by dG the least cardinal a such that G is not adistributive.Let A be an /-ideal of G.It is proved that dG and d^GJA) are mutually independent in a rather strong sense (Thm.4.2); in particular, if (XQ, otj[, • • «5 ^fi are any regular cardinals, then there exist an /-group G and /-ideals A^ a A2 cz ... cz A" of G such that dG = ag, d^GJA^ = a^ [i = 1, ..., n).The distributive radical D{G) of an /-group G is defined to be the intersection of the closures of the minimal prime subgroups of G. D{G) is a convex /-subgroup of G.The /-group G is completely distributive if and only if D(G) ~ {0} [3].We prove that if G is complete then G is the direct product M{G) X D{G), where M{G) is the greatest convex completely distributive /-subgroup of G. Some other types of convex /-subgroups /(G) of G wath the property that G is completely distributive if and only if /(G) = {0} are investigated.Let us recall some basic definitions.For lattices and lattice ordered groups (/groups) we shall use the standard notations, cf.[l], [7].The group operation will be written additively, but it is not assumed to be commutative.Let a, Д.Ье cardinals and let T,S be sets, card T^ a, card iS ^ /?.A lattice Lis said to be (л, v) --(a, j^)-di s tribut ive, if the equation (0 AVx,,, = V Лх.мо ГеТ seS (peST teT