Skew-monoidal semigroup structures on simple rings with several objects
type: Item
instance of: scholarly article
title: Skew-monoidal semigroup structures on simple rings with several objects
zbMATH Open document ID: 0611.18005
DOI: 10.1007/BF02573153
author: Hans-Jürgen Hoehnke
published in: Semigroup Forum
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134829
review text: Consider the category \({\mathfrak M}(L)^{\otimes}\) defined as the category \({\mathfrak M}(L)\) of all matrices over a ring \(L=L/K\) with identity over K (objects: finite non-empty sets; arrows: matrix multiplication), enriched with the 'skew-monoidal' Kronecker product \(\otimes\). This is almost a monoidal category; what is missing is that \(a\otimes b=(1_ P\otimes b)(a\otimes 1_ U)\)- except if L is commutative - i.e., \(\otimes\) is not functorial. This is the paradigm that the author axiomatizes under the name ''skew-monoidal categories''. He then shows that every skew-monoidal simple category is of the paradigmatic form getting a Wedderburn-Artin like theorem without recourse to Mitchell's rings with several objects. Reviewer's comment: There seems to exist a trade-off between functoriality and associativity, as seen by the reviewer's definition of a tensor product that is functorial but not associative.
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 18D35
zbMATH DE Number: 3987531
zbMATH Keywords: matrices
zbMATH Keywords: Kronecker product
zbMATH Keywords: skew-monoidal categories
zbMATH Keywords: rings with several objects
MaRDI profile type: Publication
cites work: Separable categories
cites work: Q3932925
cites work: Q5654197
cites work: Rings with several objects
cites work: Separable algebroids
OpenAlex ID: W2013670377
description: scientific article; zbMATH DE number 3987531