Coproducts in the category $${{\mathcal {S}}}(B)$$ S ( B ) of Segal topological algebras, revisited
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Coproducts in the category S (B) of Segal topological algebras, revisited Mart Abel1,2
© Akadémiai Kiadó, Budapest, Hungary 2020
Abstract In this paper we generalize the result (obtained by the author earlier in the paper titled “Products and coproducts in the category S (B) of Segal topological algebras”) about the existence of the coproduct of objects of the category S (B) from finite collection of objects of S (B) to any collection of objects of S (B). Keywords Segal topological algebra · Category · Tensor algebra · Free product · Coproduct Mathematics Subject Classification 46M05 · 46H05 · 18-XX
1 Introduction The study of Segal topological algebras in their full generality was started in [1]. In [2], the category S (B) of Segal topological algebras was defined and some of its properties established. After that, several other papers, describing different properties of the category S (B) were written. Some of them are already published [2–5], others are in print or submitted. In [5], the products and coproducts of two elements of S (B) were described. In [4], the sufficient and necessary conditions were found under which the product of any family of elements of S (B) existed. In the present paper we show that the coproduct of any family of elements of S (B) exists, generalizing the ideas of [5] (the main result of this paper is the generalization of Proposition 3.2 of [5], p. 94).
The research was supported by the institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. The author is extremely grateful to his colleagues Mati Abel, Valdis Laan, Reyna María Pérez-Tiscareño and Ülo Reimaa for their valuable comments and remarks during the discussions, while preparing the manuscript of the present paper.
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Mart Abel [email protected]; [email protected]
1
School of Digital Technologies, Tallinn University, 29 Narva Str., Room A-416, 10120 Tallinn, Estonia
2
Institute of Mathematics and Statistics, University of Tartu, Narva mnt. 18, Room 4078, 51009 Tartu, Estonia
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M. Abel
2 Tensor algebra and its algebraic properties Let (Aλ )λ∈Λ be any collection of algebras over K (where K stands for either R or C), which are made disjoint by setting a = (a, λ) for every a ∈ Aλ , if Aλ ∩ (∪μ∈Λ\{λ} Aμ ) = ∅ originally. By doing that, we will have two different elements (a, λ) and (a, μ) for every λ, μ ∈ Λ with λ = μ and every a ∈ Aλ ∩ Aμ (i.e., we actually can look at the set Aλ × {λ} = {(a, λ) : a ∈ Aλ } instead of Aλ , if there were elements, which belonged to Aλ ∩ Aμ for some different λ, μ ∈ Λ). It is known in algebra, that now one can consider the tensor product of any finite subset {A1 , . . . , An } ⊂ {Aλ : λ ∈ Λ} of algebras (Aλ )λ∈Λ with n ∈ N = {1, 2, . . .} as a set ⎧ ⎫ k ⎨ ⎬ A1 ⊗ · · · ⊗ An = a j,1 ⊗ · · · ⊗ a j,n : k ∈ N, a j,i ∈ Ai for every i ∈ {1, . . . , n} . ⎩ ⎭ j=1
It is also known that this tensor product of algebras A1 , . . . , An is a linear space over K with respect to the addition and multiplication by the elements of K, which are defined by k
a j,1
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