Lattice-ordered groups generated by ordered groups and regular systems of ideals
Groupes réticulés engendrés par un groupe ordonné et systèmes réguliers d'idéaux
Résumé
Unbounded entailment relations, introduced by Paul Lorenzen (1951), are a slight variant of a notion which plays a fundamental rôle in logic (see Scott 1974) and in algebra (see Lombardi and Quitté 2015). We propose to define systems of ideals for a commutative ordered monoid G as unbounded single-conclusion entailment relations that preserve its order and are equivariant: they describe all morphisms from G to meet-semilattice-ordered monoids generated by (the image of) G. Taking an article by Lorenzen (1953) 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 (1950); we call them regular systems of ideals. In particular, the free lattice-ordered group generated by G is described by the finest regular system of ideals 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 in our framework. In fact, Lorenzen's research in algebra is motivated by the system of Dedekind ideals for the divisibility group of an integral domain R; in particular, we provide an explicit description of the lattice-ordered group granted by Wolfgang Krull's " Fundamentalsatz " if (and only if) R is integrally closed as the "regularisation" of the Dedekind system of ideals.
Mots clés
ordered group
regular system of ideals
unbounded entailment relation
morphism from an ordered group to a lattice-ordered group
Lorenzen-Clifford-Dieudonné theorem
Fundamentalsatz for integral domains
Grothendieck lattice-ordered group.
morphism from an ordered monoid to a meet-semilattice-ordered monoid
system of ideals
unbounded single-conclusion entailment relation
ordered monoid
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