# 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