Lattice-ordered groups generated by an ordered group and regular systems of ideals

Abstract

Unbounded entailment relations, introduced by Paul Lorenzen (1951), are a slight variant of a notion which plays a fundamental role in logic (Scott 1974) and in algebra (Lombardi and Quitte 2015). We call systems of ideals their single-conclusion counterpart. If they preserve the order of a commutative ordered monoid $G$ and are equivariant with respect to its law, we call them equivariant systems of ideals for $G$: they describe all morphisms from $G$ to meet-semilattice-ordered monoids generated by (the image of) $G$. Taking a 1953 article by Lorenzen as a starting point, we also describe all morphisms from a commutative ordered group $G$ to lattice-ordered groups generated by $G$ through unbounded entailment relations that preserve its order, are equivariant, and satisfy a regularity property invented by Lorenzen; we call them regular entailment relations. In particular, the free lattice-ordered group generated by $G$ is described through the finest regular entailment relation for $G$, and we provide an explicit description for it; it is order-reflecting if and only if the morphism is injective, so that the Lorenzen-Clifford-Dieudonné theorem fits into our framework. Lorenzen's research in algebra starts as an inquiry into the system of Dedekind ideals for the divisibility group of an integral domain $R$, and specifically into Wolfgang Krull's ``Fundamentalsatz'' that $R$ may be represented as an intersection of valuation rings if and only if $R$ is integrally closed: his constructive substitute for this representation is the regularisation of the system of Dedekind ideals, i.e. the lattice-ordered group generated by it when one proceeds as if its elements are comparable.