# Refinement monoid

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In [mathematics](/source/Mathematics), a **refinement monoid** is a [commutative monoid](/source/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](/source/Conical_monoid)** if *x*+*y*=0 implies that *x*=*y*=0, for any elements *x*,*y* of *M*.

## Basic examples

A [join-semilattice](/source/Semilattice) with zero is a refinement monoid [if and only if](/source/If_and_only_if) it is [distributive](/source/Distributivity_(order_theory)).

Any [abelian group](/source/Abelian_group) is a refinement monoid.

The [positive cone](/source/Ordered_group) *G+* of a [partially ordered abelian group](/source/Ordered_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](/source/Ordered_group).

The *isomorphism type* of a [Boolean algebra](/source/Boolean_algebra_(structure)) *B* is the class of all Boolean algebras isomorphic to *B*. (If we want this to be a [set](/source/Set_(mathematics)), restrict to Boolean algebras of set-theoretical [rank](/source/Rank_(set_theory)) below the one of *B*.) The class of isomorphism types of Boolean algebras, endowed with the addition defined by [X]+[Y]=[X\times Y] (for any Boolean algebras *X* and *Y*, where [X] denotes the isomorphism type of *X*), is a conical refinement monoid.

## Vaught measures on Boolean algebras

For a [Boolean algebra](/source/Boolean_algebra_(structure)) *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](/source/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](/source/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](/source/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](/source/Ring_(mathematics)) (with unit) *R*, denote by FP(*R*) the class of [finitely generated](/source/Finitely_generated_module) [projective](/source/Projective_module) 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 [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 [X]+[Y]=[X\oplus Y], is a conical [commutative monoid](/source/Monoid). In addition, if *R* is [von Neumann regular](/source/Von_Neumann_regular_ring), then *V(R)* is a refinement monoid. It has the [order-unit](/source/Monoid) [R]. We say that *V(R)* encodes the *nonstable K-theory of R*.

For example, if *R* is a [division ring](/source/Division_ring), then the members of FP(*R*) are exactly the finite-dimensional right [vector spaces](/source/Vector_space) over *R*, and two vector spaces are isomorphic if and only if they have the same [dimension](/source/Dimension_(vector_space)). Hence *V(R)* is isomorphic to the monoid \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](/source/Field_(mathematics)) *F* is a finite [product of rings](/source/Product_of_rings) of the form M_n(F), the ring of all square [matrices](/source/Matrix_(mathematics)) 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](/source/Ordered_group) of a so-called *dimension group*. By definition, a dimension group is a [partially ordered abelian group](/source/Ordered_group) whose underlying order is [directed](/source/Directed_set), 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 \mathbb{Z}^n, is a dimension group. [Effros](/source/Edward_George_Effros), Handelman, and Shen proved in 1980 that dimension groups are exactly the [direct limits](/source/Direct_limit) 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](/source/Open_problem).

## References

1. Dobbertin, Hans (1983), "Refinement monoids, Vaught monoids, and Boolean algebras", *[Mathematische Annalen](/source/Mathematische_Annalen)*. **265** (4): 473–487, [doi:10.1007/BF01455948](https://doi.org/10.1007/BF01455948). [S2CID 119668249](https://api.semanticscholar.org/CorpusID:119668249)

1. Wehrung, Friedrich (1998), "Non-measurability properties of interpolation vector spaces", *[Israel Journal of Mathematics](/source/Israel_Journal_of_Mathematics)*. **103**: 177–206, [doi:10.1007/BF02762273](https://doi.org/10.1007/BF02762273)

1. Effros, Edward G.; Handelman, David E.; Shen, Chao-Liang (1980), "Dimension groups and their affine representations", *[American Journal of Mathematics](/source/American_Journal_of_Mathematics)*. **102** (2): 385–407, [doi:10.2307/2374244](https://doi.org/10.2307/2374244). [JSTOR 2374244](https://www.jstor.org/stable/2374244)

1. Grillet, Pierre Antoine (1976), "Directed colimits of free commutative semigroups", *Journal of Pure and Applied Algebra*. **9** (1): 73–87, [doi:10.1016/0022-4049(76)90007-4](https://doi.org/10.1016/0022-4049(76)90007-4)

1. Elliott, George A. (1976), "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras", *Journal of Algebra*. **38** (1): 29–44, [doi:10.1016/0021-8693(76)90242-8](https://doi.org/10.1016/0021-8693(76)90242-8)

1. Goodearl, K. R. & Handelman, D. E. (June 1986), "Tensor products of dimension groups and K_0 of unit-regular rings", *[Canadian Journal of Mathematics](/source/Canadian_Journal_of_Mathematics)*. **38** (3): 633–658, [doi:10.4153/CJM-1986-032-0](https://doi.org/10.4153/CJM-1986-032-0)

## 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](https://doi.org/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)", Vol. 343, Mathematics and its Applications, Springer, Dordrecht, pp. 249–255, [doi:10.1007/978-94-011-0443-2_20](https://doi.org/10.1007/978-94-011-0443-2_20)
- Goodearl, K. R. (1986), *Partially Ordered Abelian Groups with Interpolation*, Vol. 20, Mathematical Surveys and Monographs, 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

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