Abstract
We study the variety Var(ℕ) of lattice-ordered monoids generated by the natural numbers. In particular, we show that it contains all 2-generated positively ordered lattice-ordered monoids satisfying appropriate distributive laws. Moreover, we establish that the cancellative totally ordered members of Var(ℕ) are submonoids of ultrapowers of ℕ and can be embedded into ordered fields. In addition, the structure of ultrapowers relevant to the finitely generated case is analyzed. Finally, we provide a complete isomorphy invariant in the two-generated case.
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Wille, A.M. The Variety of Lattice-Ordered Monoids Generated by the Natural Numbers. Studia Logica 76, 275–290 (2004). https://doi.org/10.1023/B:STUD.0000032088.17345.ea
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DOI: https://doi.org/10.1023/B:STUD.0000032088.17345.ea