Monadic second-order subgroup property
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.