Basic Examples
A join-semilattice with zero is a refinement monoid if and only if it is distributive.
Any abelian group is a refinement monoid.
The positive cone G+ of a partially ordered abelian group G is a refinement monoid if and only if G is an interpolation group, the latter meaning that for any elements a0, a1, b0, b1 of G such that ai ≤ bj for all i, j<2, there exists an element x of G such that ai ≤ x ≤ bj for all i, j<2. This holds, for example, in case G is lattice-ordered.
The isomorphism type of a Boolean algebra B is the class of all Boolean algebras isomorphic to B. (If we want this to be a set, restrict to Boolean algebras of set-theoretical rank below the one of B.) The class of isomorphism types of Boolean algebras, endowed with the addition defined by (for any Boolean algebras X and Y, where denotes the isomorphism type of X), is a conical refinement monoid.
Read more about this topic: Refinement Monoid
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