Refinement monoids, equidecomposability types, and Boolean inverse semigroups
Résumé
For an action of a group G on a set Ω, preserving a ring B of subsets of Ω, the commutative monoid freely generated by elements [X], for X ∈ B, subjected to the relations [∅] = 0, [gX] = [X] (for g ∈ G), and [X Y ] = [X] + [Y ] (where denotes disjoint union), is called the monoid of equidecomposability types of elements of B, with respect to G, and denoted by Z + B/ /G. It is well known that Z + B/ /G is a conical refinement monoid. We observe, as an easy consequence of known results, that every countable conical refinement monoid appears as Z + B/ /G, and we develop the underlying algebraic theory, discussing in detail the quotients of refinement monoids by special sorts of congruences called V-congruences. Having in mind representation problems in nonstable K-theory of rings and operator algebras, we are naturally led to type monoids of Boolean inverse semigroups. Observing that those monoids are identical to monoids of equi-decomposability types, and formally similar to those appearing in nonstable K-theory of von Neumann regular rings, we investigate various similarities and differences between those theories. In the process, we prove that Boolean inverse semigroups form a congruence -permutable variety in the sense of universal algebra. We deduce from this that they encode a large number of embedding problems of (not necessarily Boolean) inverse semigroups into involutary rings and C*-algebras.
Mots clés
- locally matricial
- groupoid-measurable
- groupoid-induced
- semisimple
- AF
- lattice-ordered
- V-homomorphism
- path algebra
- equidecomposable
- group-measurable
- commutative
- monoid
- strongly separative
- Semigroup
- fork
- type monoid
- V-measure
- V-congruence
- group-induced
- enveloping monoid
- partial monoid
- refinement monoid
- additive homomorphism
- additive enveloping algebra
- bias
- Boolean
- distributive
- inverse
Domaines
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