Intersection operator

This is a binary subgroup property operator, viz an operator that takes as input two subgroup properties, and outputs one subgroup property

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Definition

Definition with symbols

Given two subgroup properties $p$ and $q$, the intersection operator applied to these properties, denoted as $p$ ∩ $q$, is the property defined as follows: $H$ has the property $p$ ∩ $q$ as a subgroup of $G$ if there are intermediate subgroups $K_{1}$ and $K_{2}$ of $G$ such that all the following hold:

• $H$ is the intersection of $K_{1}$ and $K_{2}$
• $K_{!}$ satisfies $p$ in $G$
• $K_{2}$ satisfies $p$ in $G$.

Property theory

Associativity

The intersection operator is both commutative and associative.

Quantalic nature

The intersection operator is a monotone operator in both arguments, when the properties are given the usual partial order of implication. Further, it distributes over logical disjunction, and is hence a quantalic property operator.

Identity element

The identity element for the composition operator is the property of being the improper subgroup, that is, of being the group embedded as a subgroup in itself. This is the same as the identtiy element (or neutral element) for the composition operator. Any property that is implied by this property is termed an identity-true subgroup property.

Nil element

The nil element for the intersection operator is the fallacy subgroup property, that is the subgroup property that is never satisfied.

Transiters

Since the intersection operator is commutative, associative and quantalic, it has a well-defined notion of transiter (with no left/right distinction thanks to commutativity). This is the intersection-transiter.