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Group with a finite complete rewriting system

Definition

A finitely generated group is termed a group with a finite complete rewriting system if it has a complete rewriting system (i.e., a rewriting system that is both finitely terminating and confluent) with respect to a finite generating set such that the rewriting system is finite in size, i.e., it makes use of only finitely many rewriting rules.

Note that it is possible for a group to have a finite complete rewriting system with respect to one finite generating set but not have any finite complete rewriting system with respect to a different finite generating set. We use the term group with a finite complete rewriting system if there exists at least one finite generating set for which a finite complete rewriting system exists.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finitely generated free group |FULL LIST, MORE INFO
finitely generated abelian group |FULL LIST, MORE INFO
finitely generated nilpotent group |FULL LIST, MORE INFO
finite group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group with solvable word problem
finitely generated group