Property space

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.

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 $$S$$ be a collection of objects. A property space on $$S$$ is a collection of elements $$p$$ where each $$p$$ is map from $$S$$ 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 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.

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 $$n$$ is prime (where $$n$$ is the parameter). Then, satisfaction of the property is equivalent to satisfaction of the proposition.

Partial order on the property space
Given properties $$p$$ and $$q$$ over the same property space, we say $$p$$ &le; $$ q$$ or $$p => q$$ if every object satisfying $$p$$ also satisfies $$q$$. This can be viewed both in terms of subcollections and propositions:


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

Verbally we say that $$p$$ is smaller and $$q$$ is larger, or equivalently, that $$p$$ is stronger and $$q$$ 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.

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 $$F$$ such that if $$p <= q$$ then $$<F(p) <= F(q)$$.


 * When the property operator $$F$$ is from a property space to itself, it is termed ascendant if $$p <= F(p)$$.


 * When the property operator $$F$$ is from a property space to itself, $$F$$ is termed idempotent if $$F^2 = F$$.

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.

The 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
To define this, we first need a notion of equivalence between subgroup embeddings. Two subgroup embeddings $$H_1$$ &le; $$G_1$$ and $$H_2$$ &le; $$G_2$$ are equivalent if there is an isomorphism of $$G_1$$ with $$G_2$$ under which $$H_1$$ maps to $$H_2$$.

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

The 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 $$f_1:G_1$$ &rarr; $$G_1$$ is equivalent to $$f_2:G_2$$ &rarr; $$G_2$$ if there is an isomorphism $$\phi$$ from $$G_1$$ to $$G_2$$ such that $$\phi.f_2 = f_1.\phi$$.

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.