It is known that every countable totally ordered set can be embedded into a countable dense one. We extend this result to totally ordered commutative monoids and to totally ordered commutative residuated lattices (the latter result fails in the absence of commutativity). The latter has applications to density elimination of semilinear substructural logics. In particular we obtain as a corollary a purely algebraic proof of the standard completeness of uninorm logic; the advantage over the known proof-theoretic proof and the semantical proof is that it is extremely short and transparent and all details can be verified easily using standard algebraic constructions.

众所周知，每一个可数的全序集合都可以嵌入到一个可数的稠密集合中。我们将这个结果扩展到完全有序的可交换幺半群和完全有序的可交换剩余格（后者的结果在没有交换性的情况下失败）。后者应用于半线性子结构逻辑的密度消除。特别地，我们作为推论获得了单规范逻辑的标准完备性的纯代数证明；与已知的证明理论证明和语义证明相比，它的优势在于它非常简短和透明，所有细节都可以使用标准代数构造轻松验证。