Labeled configuration spaces and group completions

Given a pair of a partial abelian monoid M and a pointed space X, let C^M(ℝ^∞, X) denote the configuration space of finite distinct points in ℝ^∞ parametrized by the partial monoid X ∧ M. In this note we will show that if M is embedded in a topological abelian group and if we put ±M = {a - b | a, b ∈ M} then the natural map C^M(ℝ^∞, X) → C ^±M(ℝ^∞, X) induced by the inclusion M ⊂ ±M is a group completion. This result can be applied to show that for any finite set M such that {0} subset of with not equal to M ⊂ ℤ, C^M(ℝ^∞, X) is weakly equivalent to the infinite loop space Ω^∞Σ^∞X if X is connected.