# Property space

This term is related to: property theory

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This article is about a general term. A list of important particular cases (instances) is available at Category:Property spaces

## Introductory part

### Origin of the concept

The concept of looking at the collection of all properties is an old one -- it has been used in different forms, for instance, Boolean logic, the study of power sets, propositional logic and so on.

### Origin of the term

The term *property space* has been introduced by Vipul Naik. The reason is largely in order to create a legitimate *space* within which properties reside.

## Definition

### Symbol-free definition

A property space on a collection of objects is a collection of *properties* over that collection. Here, each property over a collection is a map from the objects in that collection to a two-element set **(True, False)**.

Equivalently, it is something that every object in the collection either *has* (which happens in case it is mapped to True) or does not have (which happens in case it is mapped to False).

The *complete property space* on a collection of objects is the collection of all properties over that collection.

The collection of objects for which we can evaluate the property (that is, label it as true or false) is termed the **context space** of the property.

### Definition with symbols

Let be a collection of objects. A property space on is a collection of elements where each is map from to the two-element set **(True, False)**.

## Examples

I'll discuss some general examples of property spaces. For the property spaces specifically relevant to us, check out #Important property spaces for us below.

Let's take some simple examples. Consider the collection of all natural numbers, and the notion of being a **prime number**. Then, given any natural number, we can ask the question *Is this natural number prime?* and we'll always get a yes/no answer. Equivalently, we can consider the assertion *This number is prime* and get either a true or false depending on whether the number is actually prime.

Thus, the *property* of being prime is a map from the set of natural numbers to the two-element set **(True, False)**, hence it is a property over the context space of natural numbers.

Similarly, consider the collection of all triangles in the plane. Then, given a triangle, we can ask the question: is this an acute triangle? Every triangle either *is* acute or it isn't. Thus, the *property* of being acute is a property over the context space of triangles in the plane.

## Study of property spaces

### Two interpretations of property spaces

Properties can be interpreted in two ways:

- As
*subcollections*: A property can be identified with the subcollection comprising those elements that satisfy the property. Thus, for instance, the property of being prime can be identified with the set of prime numbers, the property of being acute-angled can be identified with the set of acute-angled triangles. In this identification, the question of whether or not a given element satisfies the property reduces to the question of whether or not a given element belongs to the subcollection. - As
*propositions*: A property can be identified with a proposition with one parameter quantified over the collection. The proposition is*true*for a particular object if and only if the object satisfies the corresponding property. Thus, for instance, the property of being prime can be identified with a one-parameter proposition of the form: the given number is prime (where is the parameter). Then, satisfaction of the property is equivalent to satisfaction of the proposition.

### Partial order on the property space

Given properties and over the same property space, we say ≤ or if every object satisfying also satisfies . This can be viewed both in terms of subcollections and propositions:

- As
*subcollections*: The subcollection satisfying is contained in the subcollection satisfying . - As
*propositions*: The proposition of sastisfying is*stronger*than, or*logically implies*, the proposition of satisfying .

Verbally we say that is smaller and is larger, or equivalently, that is stronger and is weaker.

### Tautology and fallacy

The *tautology* is a property that is satisfied by *every* element. The *fallacy* is the property not satisfied by any element.

In the partial order, the tautology is the *largest* or *weakest* property, and the fallacy is the *smallest* or *strongest* property.

### Conjunction, disjunction and negation

The conjunction of a family of properties is defined as the property of an object satisfying *all* the given properties. Thus the conjunction of the properties of being prime and odd is the property of being an odd prime, while the conjunction of the properties of being divisible by 2,3, and 5 is the property of being divisible by 30.

- As
*subcollections*: The conjunction of a family of properties corresponds to the*intersection*of the corresponding subcollections. - As
*propositions*: The conjunction of a family of properties corresponds to the logical conjunction of the corresponding propositions.

The disjunction of a family of properties is defined as the property of an object satisfying *at least one* of the given properties.

- As
*subcollections*: The disjunction of a family of properties corresponds to the*union*of the corresponding subcollections. - As
*propositions*: The disjunction of a family of properties corresponds to the*logical disjunction*of the corresponding propositions.

The negation of a property is the property of *not satisfying* that property.

- As
*subcollections*: The negation of a property corresponds to the complement of the corresponding subcollection. - As
*propositions*: The negation of a property corresponds to the logical negation of the corresponding proposition.

## Further study of property spaces

### Property operators and property modifiers

A property operator is a map from one property space to another. Some properties on which we judge property operators themselves:

- A monotone property operator is a property operator such that if then .

- When the property operator is from a property space to itself, it is termed ascendant if .

- When the property operator is from a property space to itself, is termed idempotent if .

Property operators from a property space to itself are also termed property modifiers.

### Binary operators

There is a general theory of quantalic binary property operators which arise through an *existential* definition. Most of the binary property operators that we shall be defined existentially, hence they are quantalic binary property operators.

### Metaproperties

A metaproperty is a property whose *collection* (viz, context space) is itself a property space. That is, it is a map from a property space to the two-element set **(True, False)**.

Typically, metaproperties arise through property modifiers. For instance, given a property modifier, there arises the metaproperty of being a *fixed point* under that modifier, and of being an *image point* for that modifier.

Extending the idea further, a metametaproperty is a property whose context space is itself a metaproperty space.

## Important property spaces for us

### The group property space

`Further information: Group property space`

This is the complete property space over the collection of *isomorphism class of groups*. Thus, the elements of the group property space are maps from the collection of isomorphism classes of groups to the two element set **(True, False)**.

Typical elements of the group property space include properties like being simple, finite, solvable and so on.

### The subgroup property space

`Further information: Subgroup property space`

To define this, we first need a notion of *equivalence* between subgroup embeddings. Two subgroup embeddings ≤ and ≤ are equivalent if there is an isomorphism of with under which maps to .

The subgroup property space is the complete property space over the collection of equivalence classes of subgroups.

### The function property space

`Further information: Function property space`

The function property space over groups is the complete property space over the collection of all functions from a group to itself, upto a suitable notion of *equivalence* as follows: a function → is equivalent to → if there is an isomorphism from to such that .

If we restrict the functions to only automorphisms, we get the automorphism property space, and if we restrict to only endomorphisms, we get the endomorphism property space.

## External links

- Property Theory: An Introduction provides a detailed introduction to the ideas of property theory, including full definitions and lots of examples drawn from different contexts