Balanced subgroup property (generic notion)

This article is about the general notion of balanced subgroup property. For the typical use of balanced subgroup property in the context of the function restriction formalism, refer balanced subgroup property (function restriction formalism).

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Balanced subgroup property (generic notion), all facts related to Balanced subgroup property (generic notion)) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

Definition

Symbol-free definition

We need to have two inputs with respect to which we can talk of balanced subgroup property:

• A relational mapping system
• A specification of one of these formalisms based on the relational mapping system: the restriction formalism, the extension formalism, the [[[universal restriction formalism]], the universal extension formalism, the when-defined restriction formalism, the when-defined extension formalism

For a given formalism based on the relational mapping system, a subgroup property is termed balanced if it has a formal expression (in that formalism) with equal left and right sides.

Definition with symbols

We need to have two inputs with respect to which we can talk of balanced subgroup property:

• A relational mapping system
• A specification of one of these formalisms based on the relational mapping system: the restriction formalism, the extension formalism, the [[[universal restriction formalism]], the universal extension formalism, the when-defined restriction formalism, the when-defined extension formalism

For a given formalism based on the relational mapping system, a subgroup property is termed balanced if it has a formal expression (in that formalism) of the form $a$ → $a$.