Left linear theories -- a generalization of module theory (Q1330905)

For a ring \(R\) with unit, let \(R^{(\mathbb{N})}\) be the set of all \(R\)- sequences with finite support, and let \(\delta^ j_ * = (\delta^ j_ 1, \delta^ j_ 2, \dots) \in R^{(\mathbb{N})}\), where \(\delta^ j_ k\) be the familiar Kronecker symbol \((j,k \in \mathbb{N})\). Each \(r_ * \in R^{(\mathbb{N})}\) induces an operation \(\sigma (r_ *) : (R^{(\mathbb{N})})^ \mathbb{N} \to R^{(\mathbb{N})}\) defined by \[ \sigma (r_ *) (s^ 1_ *, s^ 2_ *, \dots) = \sum_{i \in \mathbb{N}} r_ i s^ i_ *. \] The exponential transpose \(R^{(\mathbb{N})} \times (R^{(\mathbb{N})})^ \mathbb{N} \to R^{(\mathbb{N})}\) of the corresponding map \(\sigma\) now can be lifted to a multiplication \((R^{(\mathbb{N})})^ \mathbb{N} \times (R^{(\mathbb{N})})^ \mathbb{N} \to (R^{(\mathbb{N})})^ \mathbb{N}\), which together with the unit \(\delta = (\delta^ 1_ *, \delta^ 2_ *, \dots)\) comprises a monoid structure on \((R^{(\mathbb{N})})^ \mathbb{N}\). The existence, for a fixed set \(X\), of a bijective correspondence between left-\(R\)-module structures \(R \times X \to X\) on \(X\) and those left- \((R^{(\mathbb{N})} )^ \mathbb{N}\)-module structures \((R^{(\mathbb{N})})^ \mathbb{N} \times X^ \mathbb{N} \to X^ \mathbb{N}\) on \(X^ \mathbb{N}\) that arise from maps \(R^{(\mathbb{N})} \times X^ \mathbb{N} \to X\) motivates this paper. Instead of rings with unit, the authors introduce functions \({\mathcal L} \times {\mathcal L}^ N \to {\mathcal L}\), so-called left linear theories, as basis for modules. Here \(N\) is a fixed set with more than two elements intended to replace \(\mathbb{N}\), the arbitrary set \({\mathcal L}\) plays the role of \(R^{(\mathbb{N})}\), and the function, expressed by concatenation, has to satisfy \((l\lambda) \mu = l (\lambda \mu)\) for any \(l \in {\mathcal L}\) and all \(\lambda, \mu \in {\mathcal L}^ N\). (Here \(\lambda \mu\) has to be interpreted by lifting the original function to \({\mathcal L}^ N \times {\mathcal L}^ N \to {\mathcal L}^ N\).) Unital left linear theories have in addition a distinguished element \(\delta \in {\mathcal L}^ N\) that satisfies \(l \delta = l\) for all \(l \in {\mathcal L}\), and \(\delta (i) \lambda = \lambda (i)\) for every \(\lambda \in {\mathcal L}^ N\) and each \(i \in N\). This notion of left linear theory is flexible enough to encompass, e.g., the barycentric calculus, and various convexity theories, which particularly the second author has an active interest in. Left linear theories as well as (unital) left-, right-, and bimodules over fixed left linear theories can readily be organized into nice algebraic categories. The paper contains explicit descriptions of the free left-\({\mathcal L}\)- module, right-\({\mathcal M}\)-module, and \({\mathcal L}\)-\({\mathcal M}\)-bimodule over a given set. In the (bi-) category of all bimodules, the authors define a (non- symmetric) tensor product by means of a universal property with respect to ``bilinear'' maps. [Reviewer's remark: The notion of bilinearity chosen here when restricted to the variety of \({\mathcal L}\)-bimodules for fixed \({\mathcal L}\) does not reduce to the usual one for varieties, cf., e.g., \textit{B. A. Davey} and \textit{G. Davis}, Algebra Univers. 21, 68-88 (1985; Zbl 0604.08004). In fact, it is not at all clear that in this variety the resulting tensor product is symmetric. Furthermore, the category of \({\mathcal L}\)-\({\mathcal M}\)-bimodules is a variety as well, hence carries a standard notion of bilinearity. The authors' version cannot even be restricted to this variety.] It is then shown in detail that the (bi-) category of bimodules is monoidal right-closed with respect to this tensor product, however, the associativity of the tensor product is left as an open problem. The natural candidate for the associativity morphism turns out to be surjective; the authors even conjecture that in general it may not be an isomorphism. The equally interesting question (that would also settle the associativity issue), whether the (bi-) category of bimodules is monoidal left-closed, is not addressed.