Refinement monoids, equidecomposability types, and Boolean inverse semigroups

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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: group-induced; groupoid-induced; group-measurable; V-congruence; enveloping monoid; partial monoid; refinement monoid; additive enveloping algebra; additive homomorphism; bias; Boolean; distributive; inverse; commutative; monoid; Semigroup; equidecomposable; type monoid; V-measure; fork; strongly separative; path algebra; lattice-ordered; semisimple; AF; locally matricial; groupoid-measurable; V-homomorphism

Domaines: Mathématiques générales [math.GM]

Friedrich Wehrung. Refinement monoids, equidecomposability types, and Boolean inverse semigroups. 2016. ⟨hal-01197354v2⟩