Injective positively ordered monoids. I
A positively ordered monoid is a structure \((A,+,0,\leq)\), where \((A,+,0)\) is a commutative monoid and \(\leq\) is a partial preordering of \(A\) in which every element is positive and the ordering is compatible with \(+\). This work defines various kinds of complete positively ordered monoids which are contexts for studying measurable function spaces, equidecomposability types of spaces, partially ordered Abelian groups, and cardinal algebras. The strongest notion of complete, termed injective, is arithmetically determined to be complete with the added requirement of divisibility by all nonzero natural numbers. The motivation for many of the results lies in the work of \textit{A. Tarski} [Fundam. Math. 31, 47-66 (1938; Zbl 0019.05403); Cardinal algebras (1949; Zbl 0041.345)] on the existence of a homomorphism from a commutative monoid to the extended positive real line and on cardinal algebras. The first result, the injectivity of the extended positive real line, is framed in the category of positively ordered monoids, and injective positively ordered monoids are posed as an algebraic enrichment of cardinal algebras.
scientific article; zbMATH DE number 166152
scientific article