Rocky Mountain Journal of Mathematics

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

Thierry Coquand, Henri Lombardi, and Stefan Neuwirth

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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.

Article information

Rocky Mountain J. Math., Volume 49, Number 5 (2019), 1449-1489.

First available in Project Euclid: 19 September 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40]
Secondary: 06F05: Ordered semigroups and monoids [See also 20Mxx] 13A15: Ideals; multiplicative ideal theory 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.)

ordered monoid system of ideals equivariant system of ideals morphism from an ordered monoid to a meet-semilattice-ordered monoid ordered group unbounded entailment relation regular entailment relation regular system of ideals morphism from an ordered group to a lattice-ordered group Lorenzen-Clifford-Dieudonné theorem Fundamentalsatz for integral domains Grothendieck $\ell$-group cancellativity


Coquand, Thierry; Lombardi, Henri; Neuwirth, Stefan. Lattice-ordered groups generated by an ordered group and regular systems of ideals. Rocky Mountain J. Math. 49 (2019), no. 5, 1449--1489. doi:10.1216/RMJ-2019-49-5-1449.

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