The Hahn embedding theorem for a class of residuated semigroups. (English) Zbl 1484.06060
Stud. Log. 108, No. 6, 1161-1206 (2020); correction ibid. 109, No. 4, 887-901 (2021).
Summary: Hahn’s embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn’s theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial lexicographic product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains which possess only finitely many idempotent elements, by means of linearly ordered abelian groups and the partial lexicographic product construction is presented.
MSC:
| 06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |
| 06F05 | Ordered semigroups and monoids |
| 06D75 | Other generalizations of distributive lattices |
Keywords:
involutive residuated lattices; construction; representation; abelian groups; Hahn-type embeddingReferences:
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