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...s also grateful to the referee for many valuable comments. 2 Configuration space and partial abelian monoid The notion of the configuration space with summable labels appear in several papers [7],[4],=-=[9]-=-. In this section, we introduce one form of such notion adapted to the purpose of this paper. Remark that our definition of partial abelian monoid is a special case of that given in [9], of which we s...

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...s also grateful to the referee for many valuable comments. 2 Configuration space and partial abelian monoid The notion of the configuration space with summable labels appear in several papers [7],[4],=-=[9]-=-. In this section, we introduce one form of such notion adapted to the purpose of this paper. Remark that our definition of partial abelian monoid is a special case of that given in [9], of which we s...

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... labels. We utilize this to construct a machinery which produces equivariant generalized homology theories from such simple and abundant data as partial monoids. 55N20, 55N91; 55P47 1 Introduction In =-=[6]-=- we attached to any pair of a Euclidean space V and a partial abelian monoid M a space C(V, M) whose points are pairs consisting of a finite subset c of V and a map a : c → M, but (c, a) is identified...

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...at provides a natural extension of the previous constructions is to use configurations with labels in a partial abelian monoid. These constructions were introduced in different terms in both [11] and =-=[25]-=-. Forerunners appear in [2] in the form of coloured configurations, and in [9] in connection with rational curves into toric varieties. A generalization to the non-abelian case appears in [21] (see se...

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...figuration space of finite points in R∞ with labels in a partial abelian monoid M . Then ΩCΣX∧M(R∞) is a linear enriched functor of X, and hence defines a connective homology theory. We have shown in =-=[5]-=- that the classical homology, the stable homotopy, and the connective K-homology theories can be obtained in this way. If T : C0 → NG0 is an enriched functor then for every X ∈ C0 the spaces T (ΣnX) t...

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...and Jj = ∅ or xj = ∗. We define In(X) = Cn−1(I1(X)). In the above, I1(X) is considered as a partial monoid by superimposition and Cn−1(I1(X)) is the configuration space with partially summable labels.=-=[6]-=- As for Cn(X), In(X) is too small to define a map to ΩnΣnX, and we need the thickening of In(X). We fix a real number δ > 0. For any ε < δ, we denote by I1(X)ε the subspace of I1(X) consisting of ε-se...