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I'm interested in expository material, in the form of books, chapters in books, articles, blogposts, etc., about partially-ordered, and lattice-ordered semigroups and monoids.

I have found two excellent textbooks

  • Partially Ordered Algebraic Systems by László Fuchs (Pergamon Press, 1963)
  • Lattice Theory by Garrett Birkhoff, 3rd ed. (AMS, 1973)

And a more recent textbook

  • Lattices and Ordered Algebraic Structures by Thomas Scott Blyth (Springer, 2005)

I find Blyth's textbook harder to understand than the former two, and additionally it doesn't present all the results that can be found in the former two.

My only issue with Fuchs' and Birkhoff's books is that since they were published more than 45 years ago, I'm concerned that the terminology and possibly some nuances of the definitions and theorems may have changed in the meanwhile.

I'd like to point out expressly, that I am not interested in partially-ordered, and lattice-ordered groups; only in semigroups and monoids (also possibly semi-rings and semi-modules). I'm also particularly interested in the case when the underlying order space is a semi-lattice, rather than a lattice.

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    $\begingroup$ Although it's not the main subject, the following book includes some aspects of some lattice or partial ordered algebras (pogroupoids, pomonoids and others): Residuated Lattices: An algebraic Glimpse at Substructural Logics, by Galatos, Jipsen, Kowalski and Ono. From the title is more or less evident the main topic of the book. $\endgroup$ – amrsa Jun 7 '18 at 13:20
  • $\begingroup$ Both Fuchs and Birkhoff mention the following book about lattice-ordered semigroups. Leçons sur la Théorie des Treillis des Structures Algébriques Ordonnées et des Treillis Géométriques, Gauthier-Villars, Paris, 1953 $\endgroup$ – Evan Aad Jun 7 '18 at 13:52

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