Stratisfied semigroups
type: Item
instance of: scholarly article
title: Stratisfied semigroups
zbMATH Open document ID: 0822.20059
DOI: 10.1007/BF02573503
author: Pierre Antoine Grillet
published in: Semigroup Forum
publication date: 1995-02-01T00:00:00Z
full work available at URL: https://eudml.org/doc/135365
review text: A semigroup \(S\) with zero 0 is called stratisfied if \(\bigcap_{m > 0} S^ m = \{0\}\). (The definition is effective for any semigroup, that is, a semigroup \(S\) is stratisfied if \(S^ 0\) is stratisfied.) The depth function \(\lambda : S \setminus \{0\} \to N\) assigns to each \(s \in S\), \(s\neq 0\), the greatest positive integer \(m\) such that \(s \in S^ m\). A semigroup \(S\) is called homogeneous if it has a presentation in which the defining relations are of the form either \(u = 0\) or \(u = v\) with words \(u\) and \(v\) of equal lengths. A product \(a_ 1 a_ 2 \cdots a_ r\) is called proper if \(a_ 1 a_ 2 \cdots a_ r \neq 0\) and \(\lambda(a_ 1 a_ 2 \cdots a_ r) = \lambda a_ 1 + \lambda a_ 2 + \cdots + \lambda a_ r\). Then \(S\) is homogeneous if and only if \(S\) is stratisfied and all non-zero products in \(S\) are proper. The depth of a subset \(C \supsetneqq \{0\}\) of \(S\) is defined by \(\mu(C) = \max \{\lambda a: a\neq 0,\;a \in C\}\). The following are basic principles of this paper. (1) Let \(\rho\) be a congruence on a stratisfied semigroup. \(S/\rho\) is stratisfied if and only if every non-zero \(\rho\)- class has finite depth (the depth of a non-zero \(\rho\)-class \(C\) in \(S/\rho\) coincides with its depth as a subset of \(S\)). (2) Let \(S\) be homogeneous. If \(\rho\) is homogeneous, that is, every non-zero \(\rho\)- class is contained in a single layer (i.e. one of \(S^ m\setminus S^{m+1}\)), then \(S/ \rho\) is homogeneous. If \(S/ \rho\) is homogeneous and the \(\rho\)-class of every \(x \in S\setminus S^ 2\) has depth 1 then \(\rho\) is homogeneous. The class of stratisfied semigroups contains free semigroups, free commutative semigroups, homogeneous semigroups and nilpotent semigroups. Let \(\mathcal V\) be the variety containing all commutative semigroups. Let \(F\) be a free \(\mathcal V\)-semigroup and consider a homogeneous equivalence relation (h.e.r.) of degree \(m\) on \(F\), that is, an equivalence relation on \((F^ m \set…
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 716384
zbMATH Keywords: nilsemigroups
zbMATH Keywords: depth
zbMATH Keywords: presentations
zbMATH Keywords: defining relations
zbMATH Keywords: congruences
zbMATH Keywords: layers
zbMATH Keywords: stratisfied semigroups
zbMATH Keywords: free semigroups
zbMATH Keywords: free commutative semigroups
zbMATH Keywords: homogeneous semigroups
zbMATH Keywords: nilpotent semigroups
zbMATH Keywords: variety
zbMATH Keywords: free \(\mathcal V\)-semigroups
zbMATH Keywords: homogeneous equivalence relation
zbMATH Keywords: traces
reviewed by: Takayuki Tamura
MaRDI profile type: Publication
cites work: Q3848243
cites work: Isomorphisms of stratified semigroup algebras
cites work: The semigroup of one-to-one transformations with finite defects
cites work: Homomorphisms and subdirect decompositions of semigroups
cites work: Construction of Trees and Commutative Archimedean Semigroups
cites work: Construction of finite commutative $z$-semigroups
Wikidata QID: Q126270976
OpenAlex ID: W3189775397
description: scientific article; zbMATH DE number 716384