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By a lattice-ordered monoid, I mean a structure $(A,0,{+},{\vee},{\wedge})$ such that $(A,0,{+})$ is a (not necessarily commutative) monoid, $(A,{\vee},{\wedge})$ is a lattice, and the two distributive laws $$(a+x+b)\wedge(a+y+b) = a + (x\wedge y) + b$$ and $$(a+x+b)\vee(a+y+b) = a + (x\vee y) + b$$ both hold. A lattice-ordered group is defined similarly.

It is well known that the lattice reduct $(A,{\vee},{\wedge})$ of a lattice-ordered group is always distributive, but this result does not extend to lattice-ordered monoids.

Are there quasi-equational conditions on the monoid $(A,0,{+})$ that guarantee that the lattice reduct $(A,{\vee},{\wedge})$ is distributive? I am particularly interested in the case where the monoid is commutative.

I'm not looking for an exact characterization (which is probably impossible in terms of quasi-identities) but for sufficient conditions.

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Any conditions on the monoid? Should be cancelative, for example? – Steve Richards Nov 13 '10 at 21:07
@Steve: Conditions on the monoid are what I'm looking for. I'm open to any suggestions. – François G. Dorais Nov 13 '10 at 21:44

It is highly tempting to expand the signature just a bit so that we can talk about, e.g., residuated lattices. Thus, the discussion in this paper would seem generally relevant; I direct your attention particularly to theorem 15 (page 15) and to the remark near the top of page 20 that the algebras of the quasi-variety described in theorem 15 have distributive lattices as their underlying lattices. So even though such structures go outside the strict confines of your question, perhaps they give some food for thought.

(Or, perhaps you already knew of this paper?)

Just a remark that residuated lattices are important in categorical logic (cf. the concept of quantale) and have nice varietal properties, e.g., they form a Mal'cev variety.

I'll add that I have no particular expertise here, but the question looked interesting.

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Thanks Todd! I wasn't aware of that paper. – François G. Dorais Nov 14 '10 at 15:36
Sounds interesting, but Todd, that link no longer works. Can you provide another? – goblin Jan 29 at 4:47
@goblin Not sure I can (I don't remember the paper), but it looks very close to some work by one of Nikolaos Galatos, Constantine Tsinakis, or Francesco Paoli. See for example generalized MV-algebras in, or the discussion of Birkhoff's common abstraction problem in… Sorry I don't have this to hand anymore. – Todd Trimble Jan 29 at 11:22
That's fine; thanks for the links. – goblin Jan 29 at 16:27

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