Group with a finite complete rewriting system
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Contents
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 |