# Intersection operator

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

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

## Definition

### Definition with symbols

Given two subgroup properties and , the intersection operator applied to these properties, denoted as ∩ , is the property defined as follows: has the property ∩ as a subgroup of if there are intermediate subgroups and of such that all the following hold:

- is the intersection of and
- satisfies in
- satisfies in .

## 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.