Elementary algebraic subset

This article defines a property of subsets of groups
View other properties of subsets of groups|View properties of subsets of abelian groups|View subgroup properties

Definition

A subset $S$ of a group $G$ is termed an elementary algebraic subgroup if there exists a word map $w:G^{n}\to G$ and elements $g_{1},g_{2},\dots ,g_{n-1}$ such that:

$S=\{x\in G\mid w(g_{1},g_{2},\dots ,g_{n-1},x)={\mbox{identity element of }}G\}$

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
algebraic subset	arbitrary intersection of finite unions of elementary algebraic subsets			|FULL LIST, MORE INFO
unconditionally closed subset	closed for every topology making the group a T0 topological group			|FULL LIST, MORE INFO