Monadic second-order characteristic subgroup: Difference between revisions

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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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| [[Weaker than::Elementarily characteristic subgroup]] || no other elementarily equivalent subgroup || || ||
| [[Weaker than::elementarily characteristic subgroup]] || no other elementarily equivalent subgroup || || ||
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| [[Weaker than::Monadic second-order purely definable subgroup]] || can be defined using the pure theory of the group in monadic second-order language || || ||
| [[Weaker than::monadic second-order purely definable subgroup]] || can be defined using the pure theory of the group in monadic second-order language || || ||
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| [[Weaker than::Purely definable subgroup]] || can be defined purely using the first-order pure theory of the group || || ||
| [[Weaker than::purely definable subgroup]] || can be defined purely using the first-order pure theory of the group || || ||
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Latest revision as of 14:25, 1 June 2020

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

A subgroup of a group is termed a monadic second-order characteristic subgroup if there is no other subgroup of such that the monadic second-order theories of the group-subgroup pairs and coincide. In other words, can be distinguished from any other subgroup of using monadic second-order logic in the pure theory of the group .

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
elementarily characteristic subgroup no other elementarily equivalent subgroup
monadic second-order purely definable subgroup can be defined using the pure theory of the group in monadic second-order language
purely definable subgroup can be defined purely using the first-order pure theory of the group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Second-order characteristic subgroup
Characteristic subgroup
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