Elementary algebraic subset

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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