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
Famous quotes containing the words basic and/or examples:
“Not many appreciate the ultimate power and potential usefulness of basic knowledge accumulated by obscure, unseen investigators who, in a lifetime of intensive study, may never see any practical use for their findings but who go on seeking answers to the unknown without thought of financial or practical gain.”
—Eugenie Clark (b. 1922)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)