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# MATHEMATICAL STRUCTURES

## Basic idea

A mathematical structure is a set (or sometimes several sets) with various associated mathematical objects such as subsets, sets of subsets, operations and relations, all of which must satisfy various requirements (axioms). The collection of associated mathematical objects is called the structure and the set is called the underlying set.   The axiomatic method essentially always produces a mathematical structure.

### Warning

This definition of mathematical structure is not a mathematical definition. A proper mathematical definition of "mathematical structure" is best done using category theory.   The phrase “mathematical structure” is usually used in the definition or discussion of a particular kind of mathematical structure, without any general definition being given.

## Examples

¨  An equivalence relation is a set together with a relation on the set that is reflexive, symmetric and transitive.

¨  A partition is a set together with a set of subsets with the property that every element of the set is in exactly one of the subsets.

¨  A group is a set together with a binary operation with the properties that the operation is associative, there is an identity element, and every element of the set has an inverse.

¨  A topological space is a set $S$ together with a set $T$ of subsets of $S$ containing the empty set and $S$ and closed under finite intersections and all unions.  (MW, Wi).

## Terminology

In some parts of math, a math structure and its underlying set may be denoted by convention by the same symbol.  For example, if $G$ is a group, then one refers to an element $g\in G$. In that phrase, $G$ must denote the underlying set, not the whole structure

This way of speaking does not occur, for example, with structures such as equivalence relations.

## Remarks

¨  The same set can have many different structures on it, even of the same type.  The example here shows that a two-element set has two different (and non-isomorphic) partition structures and two different (but isomorphic) group structures.

¨  Widely-used mathematical objects generally have “canonical structures” of various types on them. For example, the set of integers can be ordered in many ways, but it has a particular ordering (the familiar one) that is referred to as “the ordering of the integers”.

¨  The word "structure", sometimes in the phrase "mathematical structure", is also used to describe the way certain types of mathematical objects are related to each other in a system. For example, you might investigate the structure of the solutions of a particular type of differential equation.   This is vaguely related to the kind of “mathematical structure” discussed in this section but is not the same thing.

# How to think about mathematical structures

## Minimality

Presenting a complex mathematical idea as a mathematical structure involves finding a minimal set of associated objects (the structure) and a minimal set of conditions on those objects from which the theorems about the structure follow.  The ingredients of the structure are minimal so that it is easier to verify that some object is an example of that kind of structure.   This is essentially the main use of the axiomatic method.

This minimal set of objects and conditions may not be the most important aspects of the structure for applications or for one's mental representation of the structure. See definition and function for more discussion of this.

## Different definitions for the same structure

The same kind of structure can often be defined by two or more very different kinds of minimal ingredients.   A mathematical structure of a given type has lots of structure implied by the minimalist definition, and when you think of a structure it is best to think of it as containing all that information, not just the stuff in the definition.

### Examples

¨  “Equivalence relation” and “partition” are two different ways of defining exactly the same structure on a set.  This phenomenon is discussed at length under equivalence relation.

¨  Another example is given more briefly under symmetric relation.

¨  The real numbers have many different equivalent definitions and constructions.  See the Wikipedia article.