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.

中文翻译：

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一类残差半群的哈恩嵌入定理

哈恩嵌入定理断言线性有序的阿贝尔群嵌入在实群的某些字典积中。本文将哈恩定理推广到一类残差半群，即仅具有有限多个幂等元的奇对合交换残差链。为此，引入部分字典积构造，从一对奇对合可交换剩余格构造新的奇对合可交换剩余格，以及仅具有有限多个幂等元素的奇对合可交换剩余链的表示定理，提出了线性有序阿贝尔群和部分词典产品构造。