# Intersection operator

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

## Contents

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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.