Monadic second-order subgroup property

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This article defines a subgroup metaproperty in terms of the model-theoretic approach: viz by looking at the language in which the subgroup property can be defined

This article is about a general term. A list of important particular cases (instances) is available at Category:Monadic second-order subgroup properties

Definition

Symbol-free definition

A subgroup property is termed a monadic second-order group property if it can be expressed via monadic second-order logic; in other words, it can be expressed in a language which allows for:

  • Logical operations (conjunction, disjunction, negation, and conditionals)
  • Equality testing
  • Quantification over elements and subsets of the group and subgroup (in particular, this allows one to test membership of an element in the group, in the subgroup)
  • Group operations (multiplication, inverse element and identity element)

Since we can quantify over all subsets, and can test closure under group operations, we can also quantify over all subgroups.

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties