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Unbounded entailment relations, introduced by Paul Lorenzen (1951), are aslight variant of a notion which plays a fundamental r\^ole in logic (see Scott1974) and in algebra (see Lombardi and Quitt\'e 2015). We call systems ofideals their single-conclusion counterpart. If they preserve the order of acommutative ordered monoid G and are equivariant w.r.t. its law, we call themequivariant systems of ideals for G: they describe all morphisms from G tomeet-semilattice-ordered monoids generated by (the image of) G. Taking anarticle by Lorenzen (1953) as a starting point, we also describe all morphismsfrom a commutative ordered group G to lattice-ordered groups generated by Gthrough unbounded entailment relations that preserve its order, areequivariant, and satisfy a regularity property invented by Lorenzen (1950); wecall them regular entailment relations. In particular, the free lattice-orderedgroup generated by G is described through the finest regular entailmentrelation for G, and we provide an explicit description for it; it isorder-reflecting if and only if the morphism is injective, so that theLorenzen-Clifford-Dieudonn\'e theorem fits into our framework. Lorenzen'sresearch in algebra starts as an inquiry into the system of Dedekind ideals forthe divisibility group of an integral domain R, and specifically into WolfgangKrull's "Fundamentalsatz" that R may be represented as an intersection ofvaluation rings if and only if R is integrally closed: his constructivesubstitute for this representation is the regularisation of the system ofDedekind ideals, i.e. the lattice-ordered group generated by it when oneproceeds as if its elements are comparable.

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