T.i. subgroup property

Symbol-free definition
A subgroup property is termed t.i. if it is both transitive and identity-true with respect to the composition operator. That is, $$p$$ is t.i. if $$e$$ &le; $$p$$ and $$p * p$$ &le; $$p$$.

Definition with symbols
A subgroup property $$p$$ is termed t.i. if it satisfies the following two conditions:


 * For any group $$G$$, $$G$$ satisfies $$p$$ as a subgroup of itself. This is the condition of being identity-true.
 * If $$G$$ &le; $$H$$ &le; $$K$$, such that $$G$$ satisfies $$p$$ in $$H$$ and $$H$$ satisfies $$p$$ in $$K$$ then $$G$$ satisfies $$p$$ in $$K$$. This is the condition of being transitive.

Property submonoid
The natural significance of t.i. properties with respect to the composition operator arises as follows. Consider the property space of all subgroup properties, equipped with a monoid structure via the composition operator. Now take any subgroup property $$p$$. Then the map sending an arbitrary property $$q$$ to the conjunction of $$p$$ with $$q$$, is an endomorphism of the property monoid if and only if $$p$$ is a t.i. subgroup property.


 * The identity-trueness is needed to ensure that the identity element is preserved.
 * The transitivity is needed to ensure that the multiplicative structure is preserved.

Thus, conjunction with a t.i. subgroup property gives a property submonoid.

Category-theoretic interpretation
If we consider the category whose objects are groups and whose morphisms are injective group homomorphisms, then t.i. subgroup properties are precisely the properties that describe subcategories of this category.

Fixed point space of idempotent operators
The collection of t.i. subgroup properties is precisely the fixed point space of the following three idempotent subgroup operators :


 * The left transiter operator
 * The right transiter operator
 * The subordination operator