Injective positively ordered monoids I

Abstract

We define in this paper a certain notion of completeness for a wide class of commutative (pre)ordered monoids (from now on P.O.M.'s). This class seems to be the natural context for studying structures like measurable function spaces, equidecomposability types of spaces, partially ordered abelian groups and cardinal algebras. Then, we can prove that roughly speaking, spaces of measures with values in complete P.O.M.'s are complete P.O.M.'s. Furthermore, this notion of completeness yields us an 'arithmetical' characterization of injective P.O.M.'s.

Representations of distributive semilattices in ideal lattices of various algebraic structures

“…The latter result follows immediately from [35,Theorem 3.11], but it can also be proved directly. Moreover, our proof here is self-contained.…”

“…It is to be noted that in [8], every refinement monoid is, in addition, required to satisfy the axiom x + y = 0 ⇒ x = y = 0 (conicality), while this is not the case for most other authors (e.g., [1], [35]). …”

Monoids of Intervals of Simple Refinement Monoids and Non-stable K-Theory of Multiplier Algebras

“…Given (M, u) and (N, v) monoids with order-unit, a monoid morphism f : M → N is said to be normalized provided that f (u) = v. We say that M is simple if M is non-zero, conical, and every non-zero element is an order-unit. For other basic definitions and results on abelian monoids, see for example [19], [28] and [29]. Now, we recall some definitions (see, e.g.…”

Embedding rank one simple groups into rank one simple Riesz groups

“…Since most definitions about abelian monoids are analogous to those of partially ordered abelian groups, we will use them without an explicit definition. For basic definitions and results on abelian monoids, see for example [11], [16] and [17]. Let G þ be the positive cone of a partially ordered abelian G. A nonempty subset X of G þ is called an interval in G þ if X is upward directed and order-hereditary.…”