Categories of representations of a class of commutative cancellative semigroups
type: Item
instance of: scholarly article
zbMATH Open document ID: 1001.20052
author: Mario Petrich
author: Antonio M. Cegarra
publication date: 2002-12-16T00:00:00Z
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20M30
zbMATH DE Number: 1709126
zbMATH Keywords: commutative semigroups
zbMATH Keywords: commutative cancellative idempotent free subarchimedean semigroups
zbMATH Keywords: \(\mathcal P\)-semigroups
zbMATH Keywords: categories
zbMATH Keywords: pivot elements
zbMATH Keywords: equivalences of categories
zbMATH Keywords: representative embeddings
zbMATH Keywords: functors
MaRDI profile type: Publication
title: Categories of representations of a class of commutative cancellative semigroups
published in: Algebra Colloquium
review text: Let \(S\) be a commutative semigroup. \(S\) is called subarchimedean if there exists \(z\in S\) such that for any \(a\in S\) there are \(n>0\) and \(x\in S\) such that \(z^n=ax\). A commutative cancellative idempotent free subarchimedean semigroup is called a \(\mathcal P\)-semigroup. (The reviewer and his students called these semigroups \(\overline{\mathcal N}\)-semigroups and studied them by the usual treatment [\textit{H. B. Hamilton, T. E. Nordahl, T. Tamura}, Pac. J. Math. 61, 441-456 (1975; Zbl 0358.20073)].) In this paper the authors perform a categorical treatment which nicely uniformly reorganizes and extends widely the results of the reviewer and his students.NEWLINENEWLINENEWLINEFirst, notations, \(P\) the set of positive integers, \(N\) nonnegative integers, \(Z\) integers, \(Q\) rational numbers. Let \(X\) be a set, \(\iota_X\) the identity map on \(X\), \(0_X\colon X\to 0\), \(1_X\colon X\to 1\). For \(\varphi\colon X\to Q\), define \(\varphi\pm 1\) by \(a(\varphi\pm 1)=a\varphi\pm 1\). Let \(A\), \(B\) be Abelian groups, if \(\theta\colon A\to B\), \((a,b)\theta=(a\theta)(b\theta)((ab)\theta)^{-1}\) for \(a,b\in A\). For \(I\colon G\times G\to Z\) and \(a\in G\), let \(aI_{-1}=aI_0=0\) and \(aI_n=\sum^n_{i=1}(a,a^i)I\) for \(n\in P\). In the construction below the following are assumed. NEWLINE\[NEWLINE\begin{matrix}{\l}\quad &{\l}\\ (R_N)\quad(a,b)\varphi\in N,&(R_Z)\quad(a,b)\varphi\in Z,\\ (N_1)\quad e\varphi=(e,e)I =1,&(N_0)\quad e\varphi=(e,e)I=0,\\ (C)\quad(a,b)I=(b,a)I,&(A)\quad(a,b)I+(ab,c)I=(a,bc)I+(b,c)I,\\ (M_\kappa)\quad(s,a)(t,b)=(s+t,ab),&(M_y)\quad(m,a)(n,b)=(m+n+(a,b)I,ab).\end{matrix}NEWLINE\]NEWLINE Let \(G\) be a group, \(I\colon G\times G\to N\) satisfying \((A)\), \((N_1)\); define a groupoid \(N(G,I)\) on the set \(N\times G\) by \((N_y)\). Let \(G\) be a group, \(I\colon G\times G\to Z\) with \((A)\) and \((N_0)\), define a groupoid \(Z(G,I)\) on the set \(Z\times G\) by \((M_y)\). Let \(G\) be an Abelian group, \(\varphi\colon…
reviewed by: Takayuki Tamura
description: scientific article; zbMATH DE number 1709126