In mathematics, a refinement monoid is a commutative monoid M such that for any elements a₀, a₁, b₀, b₁ of M such that a₀+a₁=b₀+b₁, there are elements c₀₀, c₀₁, c₁₀, c₁₁ of M such that a₀=c₀₀+c₀₁, a₁=c₁₀+c₁₁, b₀=c₀₀+c₁₀, and b₁=c₀₁+c₁₁.

A commutative monoid M is said to be conical if x+y=0 implies that x=y=0, for any elements x,y of M.

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 ${\displaystyle [X]+[Y]=[X\times Y]}$ (for any Boolean algebras X and Y, where ${\displaystyle [X]}$ denotes the isomorphism type of X), is a conical refinement monoid.

Vaught measures on Boolean algebras

For a Boolean algebra A and a commutative monoid M, a map μ : A → M is a measure, if μ(a)=0 if and only if a=0, and μ(a ∨ b)=μ(a)+μ(b) whenever a and b are disjoint (that is, a ∧ b=0), for any a, b in A. We say in addition that μ is a Vaught measure (after Robert Lawson Vaught), or V-measure, if for all c in A and all x,y in M such that μ(c)=x+y, there are disjoint a, b in A such that c=a ∨ b, μ(a)=x, and μ(b)=y.

An element e in a commutative monoid M is measurable (with respect to M), if there are a Boolean algebra A and a V-measure μ : A → M such that μ(1)=e---we say that μ measures e. We say that M is measurable, if any element of M is measurable (with respect to M). Of course, every measurable monoid is a conical refinement monoid.

Hans Dobbertin proved in 1983 that any conical refinement monoid with at most ℵ₁ elements is measurable.^[1] He also proved that any element in an at most countable conical refinement monoid is measured by a unique (up to isomorphism) V-measure on a unique at most countable Boolean algebra. He raised there the problem whether any conical refinement monoid is measurable. This was answered in the negative by Friedrich Wehrung in 1998.^[2] The counterexamples can have any cardinality greater than or equal to ℵ₂.

Nonstable K-theory of von Neumann regular rings

For a ring (with unit) R, denote by FP(R) the class of finitely generated projective right R-modules. Equivalently, the objects of FP(R) are the direct summands of all modules of the form Rⁿ, with n a positive integer, viewed as a right module over itself. Denote by ${\displaystyle [X]}$ the isomorphism type of an object X in FP(R). Then the set V(R) of all isomorphism types of members of FP(R), endowed with the addition defined by ${\displaystyle [X]+[Y]=[X\oplus Y]}$, is a conical commutative monoid. In addition, if R is von Neumann regular, then V(R) is a refinement monoid. It has the order-unit ${\displaystyle [R]}$. We say that V(R) encodes the nonstable K-theory of R.

For example, if R is a division ring, then the members of FP(R) are exactly the finite-dimensional right vector spaces over R, and two vector spaces are isomorphic if and only if they have the same dimension. Hence V(R) is isomorphic to the monoid ${\displaystyle \mathbb {Z} ^{+}=\{0,1,2,\dots \}}$ of all natural numbers, endowed with its usual addition.

A slightly more complicated example can be obtained as follows. A matricial algebra over a field F is a finite product of rings of the form ${\displaystyle M_{n}(F)}$, the ring of all square matrices with n rows and entries in F, for variable positive integers n. A direct limit of matricial algebras over F is a locally matricial algebra over F. Every locally matricial algebra is von Neumann regular. For any locally matricial algebra R, V(R) is the positive cone of a so-called dimension group. By definition, a dimension group is a partially ordered abelian group whose underlying order is directed, whose positive cone is a refinement monoid, and which is unperforated, the letter meaning that mx≥0 implies that x≥0, for any element x of G and any positive integer m. Any simplicial group, that is, a partially ordered abelian group of the form ${\displaystyle \mathbb {Z} ^{n}}$, is a dimension group. Effros, Handelman, and Shen proved in 1980 that dimension groups are exactly the direct limits of simplicial groups, where the transition maps are positive homomorphisms.^[3] This result had already been proved in 1976, in a slightly different form, by P. A. Grillet.^[4] Elliott proved in 1976 that the positive cone of any countable direct limit of simplicial groups is isomorphic to V(R), for some locally matricial ring R.^[5] Finally, Goodearl and Handelman proved in 1986 that the positive cone of any dimension group with at most ℵ₁ elements is isomorphic to V(R), for some locally matricial ring R (over any given field).^[6]

Wehrung proved in 1998 that there are dimension groups with order-unit whose positive cone cannot be represented as V(R), for a von Neumann regular ring R.^[2] The given examples can have any cardinality greater than or equal to ℵ₂. Whether any conical refinement monoid with at most ℵ₁ (or even ℵ₀) elements can be represented as V(R) for R von Neumann regular is an open problem.

References

Further reading

• Dobbertin, Hans (1986), "Vaught measures and their applications in lattice theory", Journal of Pure and Applied Algebra, 43 (1): 27–51, doi:10.1016/0022-4049(86)90003-4
• Goodearl, K. R. (1995), "von Neumann regular rings and direct sum decomposition problems", Abelian groups and modules (Padova, 1994), Mathematics and its Applications, vol. 343, Springer, Dordrecht, pp. 249–255, doi:10.1007/978-94-011-0443-2_20
• Goodearl, K. R. (1986), Partially Ordered Abelian Groups with Interpolation, Mathematical Surveys and Monographs, vol. 20, American Mathematical Society, Providence, RI, ISBN 0-8218-1520-2
• Goodearl, K. R. (1991), Von Neumann Regular Rings. Second edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, ISBN 0-89464-632-X
• Tarski, Alfred (1949), Cardinal Algebras. With an Appendix: Cardinal Products of Isomorphism Types, by Bjarni Jónsson and Alfred Tarski, Oxford University Press, New York