Bass-Serre theory for groupoids and the structure of full regular semigroup amalgams
type: Item
instance of: scholarly article
title: Bass-Serre theory for groupoids and the structure of full regular semigroup amalgams
zbMATH Open document ID: 0858.20055
author: Stephen Haataja
author: S. W. Margolis
author: J. C. Meakin
published in: Journal of Algebra
publication date: 1997-03-11T00:00:00Z
review text: While it has been known since 1975 that all amalgams in the category of inverse semigroups are strongly embeddable, detailed knowledge of the structure of the associated amalgamated free products has been scarce. The main purpose of \S 3 of this paper is an investigation of the maximal subgroups of the amalgamated free product \(S_1*_US_2\) of inverse semigroups \(S_1\), \(S_2\), over a common full inverse subsemigroup \(U\) (``full'' meaning that \(U\) contains all the idempotents of both \(S_1\) and \(S_2\)). In fact the general methods employed are applicable within the category of regular semigroups. That the amalgamated free product over a full common regular subsemigroup exists was shown by \textit{T. E. Hall} [Trans. Am. Math. Soc. 246, 395-406 (1978; Zbl 0401.20053)]. However, the methods in the paper under review are based on \textit{K. S. S. Nambooripad}'s association of an ``inductive groupoid'' with each regular semigroup. He and \textit{F. Pastijn} [Houston J. Math. 15, No. 2, 249-254 (1989; Zbl 0697.20048)] showed that when \(U\) is full in \(S_1\) and \(S_2\), the groupoid associated with the amalgamated free product is just the amalgamated free product of the groupoids associated with \(S_1\) and \(S_2\), over that associated with \(U\). The maximal subgroups of \(S_1*_US_2\) are the vertex groups of this groupoid. Now amalgamated free products of groupoids have been shown by E. T. Ordman to behave in a manner akin to those of groups. The major innovation of this paper is the recognition that the vertex groups in the groupoid product are the fundamental groups of a certain graph of groups, arising quite simply from the sets of \({\mathcal D}\)-classes of \(S_1\), \(S_2\) and \(U\). This involves the Bass-Serre theory of groups acting on graphs. Various applications of the general theory are given. For instance, it is shown that if both \(S_1\) and \(S_2\) are combinatorial (that is, they have trivial subgroups), then the subgroups of any amalgamated …
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 923671
zbMATH Keywords: categories of inverse semigroups
zbMATH Keywords: amalgamated free products
zbMATH Keywords: maximal subgroups
zbMATH Keywords: full inverse subsemigroups
zbMATH Keywords: category of regular semigroups
zbMATH Keywords: amalgamated free products of groupoids
zbMATH Keywords: fundamental groups
zbMATH Keywords: graphs of groups
zbMATH Keywords: Bass-Serre theory
zbMATH Keywords: groups acting on graphs
reviewed by: Peter R. Jones
MaRDI profile type: Publication
full work available at URL: jabr.1996.0206
OpenAlex ID: W1977670827
DOI: 10.1006/JABR.1996.0206
description: scientific article; zbMATH DE number 923671