From ordered groups to ordered monoids and back again

Abstract: (Joint work with Almudena Colacito, Nikolaos Galatos, and Simon Santschi)

Removing the inverse operation from any lattice-ordered group (l-group), such as the ordered additive group of integers, produces a distributive lattice-ordered monoid (l-monoid), but it is not the case that every distributive l-monoid admits a group structure. In particular, every l-group embeds into an l-group of automorphisms of some chain and is either trivial or infinite, whereas every distributive l-monoid embeds into a possibly finite l-monoid of endomorphisms of some chain.

In this talk, we will see that inverse-free abelian l-groups generate only a proper (infinitely based) subvariety of the variety of commutative distributive l-monoids, but inverse-free l-groups generate the whole variety of distributive l-monoids. We will also see that the validity of an l-group equation can be reduced to the validity of a (constructible) finite set of l-monoid equations, yielding --- since the variety of distributive l-monoids has the finite model property — an alternative proof of the decidability of the equational theory of l-groups.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic