Refinement monoid

Last updated December 03, 2023

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.

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 $[X]+[Y]=[X\times Y]$ (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 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 μmeasurese. 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 $[X]$ 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. In addition, if R is von Neumann regular, then V(R) is a refinement monoid. It has the order-unit $[R]$ . 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 $\mathbb {Z} ^{+}=\{0,1,2,\dots \}$ 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 $M_{n}(F)$ , 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 $\mathbb {Z} ^{n}$ , 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.

Related Research Articles

In mathematics, specifically in functional analysis, a C^∗-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being $0$ .

In algebra, the kernel of a homomorphism is generally the inverse image of 0. An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.

In mathematics, a monoidal category is a category $equipped with a bifunctor$

In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group, the finite abelian groups, and the additive group of the integers, the real numbers, and every finite-dimensional vector space over the reals or a $p$ -adic field.

In mathematics and functional analysis, a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators. One of von Neumann's goals in this paper was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings.

In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.

In mathematics, the Grothendieck group, or group of differences, of a commutative monoid $M$ is a certain abelian group. This abelian group is constructed from $M$ in the most universal way, in the sense that any abelian group containing a homomorphic image of $M$ will also contain a homomorphic image of the Grothendieck group of $M$ . The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation $, a unary operation, and the constant, satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasiewicz logic; the letters MV refer to the many-valued logic of Łukasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras.$

In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ₁ compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ₂ compact elements using a construction based on Kuratowski's free set theorem.

In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings of toric varieties.

In mathematics, an approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of AF algebras using the K₀ functor whose range consists of ordered abelian groups with sufficiently nice order structure.

References

↑ Dobbertin, Hans (1983), "Refinement monoids, Vaught monoids, and Boolean algebras", Mathematische Annalen , 265 (4): 473–487, doi:10.1007/BF01455948, S2CID 119668249
1 2 Wehrung, Friedrich (1998), "Non-measurability properties of interpolation vector spaces", Israel Journal of Mathematics , 103: 177–206, doi: 10.1007/BF02762273
↑ Effros, Edward G.; Handelman, David E.; Shen, Chao-Liang (1980), "Dimension groups and their affine representations", American Journal of Mathematics , 102 (2): 385–407, doi:10.2307/2374244, JSTOR 2374244
↑ 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
↑ 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
↑ Goodearl, K. R.; Handelman, D. E. (June 1986), "Tensor products of dimension groups and $K_{0}$ of unit-regular rings", Canadian Journal of Mathematics , 38 (3): 633–658, doi: 10.4153/CJM-1986-032-0