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Posted 14
July 2007
MATHEMATICAL
STRUCTURES
Contents
How
to think about 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 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.
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.