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

Artikel i vetenskaplig tidskrift
Författare Thierry Coquand H. Lombardi S. Neuwirth Rocky Mountain Journal of Mathematics 49 5 1449-1489 0035-7596 2019 Institutionen för data- och informationsteknik (GU) 1449-1489 en dx.doi.org/10.1216/rmj-2019-49-5-14... Ordered monoid, system of ideals, equivariant system of ideals, morphism from an ordered monoid, Mathematics Matematik

## Sammanfattning

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-Dieudonne 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.