Refinement Monoid - Basic Examples

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 a_i ≤ b_j for all i, j<2, there exists an element x of G such that a_i ≤ x ≤ b_j 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.