Hahn-type embedding theorem for a class of residuated chains

A structural description of absorbent-continuous group-like commutative residuated lattices over complete, order-dense chains will be presented. The theorem is sharp, no further generalization is possible. Group-like commutative residuated lattices will be characterized as Abelian lattice-ordered groups deprived of their cancellative property only. The so-called partial-lexicographic product constructions (two of them) will be introduced, which construct group-like commutative residuated lattices. As a side-effect, it gives rise to the so-called involutive ordinal sum construction, which constructs grouplike commutative residuated lattices from a family of grouplike commutative residuated lattices. Via two decomposition theorems, corresponding to the partial-lexicographic product constructions, it will be shown that any order-dense group-like commutative residuated chain, which has only a finite number of idempotents can be built by iterating finitely many times the partial-lexicographic product constructions using solely totally ordered Abelian groups, as building blocks. The result extends the famous structural description of totally ordered Abelian groups by Hahn [4], to order-dense group-like commutative residuated chains with finitely many idempotents.