Abstract
A necessary and sufficient condition is given in order that a lattice-ordered semigroup admit a totally ordered ℓ-homomorphic image (Theorem 1). This permits us to give (Theorem 2) a necessary and sufficient condition that a lattice-ordered semigroup be representable, that is, that it be ℓ-isomorphic to a subdirect product of totally ordered semigroups, thereby solving a problem posed by L. Fuchs ([2], p. 289). We also establish a necessary and sufficient condition that a semigroup be an o-semigroup, that is, that it admit a structure of totally ordered semigroup (Theorem 3).
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References
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The central results of this note were presented by M. L. Dubreil-Jacotin in an invited address at the Symposium on Semigroups and the Multiplication Structure of Rings held at the University of Puerto Rico, Mayaguëz, Puerto Rico, March 9–13, 1970.
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Merlier, T. Sur les demi-groupes reticules et les o-demi-groupes. Semigroup Forum 2, 64–70 (1971). https://doi.org/10.1007/BF02572273
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DOI: https://doi.org/10.1007/BF02572273