Bases of Pre-Riesz Groups and Conrad’s F-Condition

Abstract

Let L(S) denote the set of lower bounds of a set S in partially ordered set T and let G + denote the positive cone of a partially ordered group G. We study directed groups G with the (pR) property: ifCalling these groups pre-Riesz, we show that Conrad's F-condition which was stated for lattice-ordered groups can still be stated for pre-Riesz groups and has similar effects modulo minor changes in definitions of basic elements and bases. As applications of our work we study integral domains whose groups of divisibility and groups of * -invertible * -ideals, for finite character star operations * , are pre-Riesz and pre-Riesz satisfying Conrad's F-condition.

Riesz and pre-Riesz monoids

“…That was the reason why I looked into the bases of pre-Riesz groups via Conrad's F-condition, with Y.C. Yang [36]. I was hoping to find the ultimate building blocks of factorization, in the positive cones of pre-Riesz groups, as Conrad did in the form of a basic element in the case of lattice ordered groups.…”

“…For a start let us recall that a nonzero finitely generated ideal A of a domain D is called primitive if A ⊆ aD implies that a is a unit, for each a ∈ D and A is super-primitive if A v = D. Call D a PSP (primitive is super-primitive) domain if every primitive finitely generated ideal of D is super-primitive. It was established in [36] that D is a PSP domain if and only if G(D), the group of divisibility of D, is a pre-Riesz group. Indeed, in light of what we have established in this paper, D is a PSP domain if the monoid of nonzeero principal ideals of D is a pre-Riesz monoid.…”

“…So if (1) D is a semirigid PSP (pre-Schreier or GCD) domain, all of m(D) is v-factorial. For (2) recall from section 4 of [36] that a PSP domain…”

“…Having described the set of star ideals of a domain, at such length, we owe it to the reader to mention a recently studied generalization of Riesz groups and hence of Riesz monoids. In [36] a directed p.o. group G =< G, * , e, ≤> was called a pre-Riesz group if for e < x 1 , x 2 , ..., x n ∈ G, with L({x 1 , x 2 , ..., x n }) = L({e}), there exists r ∈ G such that e < r ≤ x 1 , x 2 , ..., x n .…”

On $\ast$-homogeneous ideals

“…This leads to the question: If D is a domain with a finite character star operation * defined on it such that every nonzero non unit x of D is contained in some * -homogeneous ideal I of D, must D be * -potent? This question came up in a different guise as: when is a certain type of domain * s -potent for a general star operation * in [42] and sort of settled in a tentative fashion in Proposition 5.12 of [42] saying, in the general terms being used here, that: Suppose that D is a domain with a finite character * -operation defined on it. Then D is * -potent provided (1) every nonzero non unit x of D is contained in some * -homogeneous ideal I of D and (2) for M, M α ∈ * -max(D), M ⊆ ∪M α implies M = M α for some α.…”