This article describes an organization principle behind Groupprops, as well as the implementation of the principle in this wiki
The property-theoretic paradigm is one of the basic organizational principles for groupprops. It is also being used in the development of the following wikis:
This article describes the property-theoretic paradigm, and how and why it is particularly important in the context of Groupprops.
Central tenet of the property-theoretic paradigm
The central tenet is:
A whole lot of theory can be expressed in terms of properties over structures and the algebraic and logical relations between those properties
While this may seem a philosophical statement, it has a fairly concrete meaning, as is elaborated below.
What is a property?
A property over a collection of objects is something which classifies those objects into haves and have nots. In other words, given an object, it should either have that property or not have that property.
Examples from elementary mathematics:
- Being prime is a property over the collection of natural numbers. In other words, every natural number either has the property of being prime, or does not have the property of being prime.
- Being positive is a property over the collection of real numbers. In other words, every real number either has the property of being positive, or does not have the property of being positive.
Examples from group theory:
- Being solvable is a property over the collection of groups. In other words, every group either has the property of being solvable, or does not have the property of being solvable (every group is either solvable or not).
- Being normal is a property over the collection of subgroups. In other words, given a group and a subgroup of the group, the subgroup either has the property of being normal, or does not have the property of being normal.
- Being inner is a property over the collection of automorphisms. In other words, given a group and an automorphism of the group, the automorphism either has the property of being inner, or does not have the property of being inner.
Further information: Property
How properties arise
Terms and contexts arise by abstracting the same thing occurring again and again. For instance, the number three may be defined as the common element between three cars, three balloons, three children, three stars, and so on.
Similarly, a property is a comon abstraction of common features that we may observe between different objects of study. In particular:
- We may notice that the objects that we are interested in do not span all kinds of objects in the collection. Rather, there is a particular property that they all satisfy.
- We may notice that if we assume our objects to satisfy a certain property, then we can deduce a lot of things about them.
Sometimes, properties arise from other properties, by varying them around, by trying to find opposites, by trying to deduce stuff from them.
Additional structure on properties
Logical structure of properties
The most important aspect of properties is the logical structure. Given two properties, we can talk of their logical conjunction (the AND of the properties), their logical disjunction (the OR of the properties). Given a single property, we may talk of its logical negation. Logical conjunction, disjunction, and negation make the collection of possible properties into a complete Boolean lattice (roughly speaking, one where we can do conjunctions, disjunctions, and negations such that distributivity laws hold).
Algebraic structure on properties
Apart from the logical structure on the collection of properties, we