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Posted 11 October 2008
MATHEMATICAL STRUCTURES

How
to think about mathematical structures
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.
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.
¨ 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).
In some parts of math, a math structure and its underlying set may be denoted by the same symbol. For example, if G is a group, then one refers to an element . 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.
¨ The same set can have many different structures on it, even
of the same type. The example here shows that a twoelement set has two
different (and nonisomorphic)
partition structures and two different (but isomorphic) group structures.
¨ Widelyused 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.
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.
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.
¨ “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.