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 \oplus Y ] = [X] + [Y ] (where \oplus denotes disjoint union), is called the monoid of equidecomposability types of elements of B, with respect to G, and denoted by Z + /G. It is well known that Z + /G is a conical refinement monoid. We observe, as an easy consequence of known results, that every countable conical refinement monoid appears as Z + /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 equidecomposability 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: semigroup; monoid; commutative; inverse; distributive; Boolean; bias; tight map; tight enveloping algebra; refinement monoid; partial monoid; enveloping monoid; type monoid; equidecomposable; V-measure; V-homomorphism; V-congruence; group-induced; groupoid-induced; group-measurable; groupoid-measurable; locally chartable; AF; semisimple; lattice-ordered; path algebra; strongly separative; fork

Domaines: Mathématiques générales [math.GM]

Friedrich Wehrung. Refinement monoids, equidecomposability types, and Boolean inverse semigroups. 2015. ⟨hal-01197354v1⟩