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Abelian group – complete ordered field
Abelian group
A group in which the group operation is commutative.
algebra
The term algebra has broadened enormously in modern mathematics from its original meaning as the abstract study of the laws of arithmetic, in which letters or other symbols are used in place of specific numbers in equations or other arithmetic statements. The term algebra should now be understood to denote any set of objects together with a collection of finitary operations defined on it. Thus, we may have an algebra of sets, an algebra of numbers, an algebra of functions, and so on. See the following entries for many particular uses of the words "algebra" and "algebaic."
algebraically closed
A field F is algebraically closed if every polynomial with coefficients in F has a root in F. Neither the field of rational numbers nor the field of real numbers is algebraically closed, but the field of complex numbers is algebraically closed.
algebraic lattice
A lattice L is said to be algebraic if every element of L is a join of compact elements of L.
algebraic number
A real or complex number which is the root of a polynomial with rational coefficients. Numbers that cannot be so expressed are called transcendental numbers. The algebraic numbers are countable.
algebra of sets
Given a set X, an algebra on X is a collection of subsets of X which is closed under finite unions and complements. Such an algebra always includes the empty set and X itself. An algebra may also be defined as a ring of sets which includes X. An algebra of sets which is closed under countable unions is called a s-algebra.
alternating group
The subgroup of a symmetric group on n elements which contains all of the even permutations is called the alternating group on n elements.
antisymmetric relation
A relation “~” on a set X is antisymmetric if, for all x and y in X, whenever x ~ y and y ~ x then x = y.
Cf. symmetric relation, asymmetric relation.
associative
An operation “ · ” on a set A is associative if for all a, b, and c in A, (a · b) · c = a · (b · c).
Cf. commutative, distributive.
associative property
A property of numbers which states that the operations of addition and multiplication are associative.
Cf. commutative, distributive.
asymmetric relation
A relation “~” on a set X is asymmetric if, for all x and y in X, x ~ y implies that it is not the case that y ~ x.
Cf. symmetric relation, antisymmetric relation.
automorphism
An isomorphism of a space to itself, that is, a bijective function on a space that preserves relationships, operations, and/or all essential properties (continuity, order, etc.).
Cf. homomorphism, group automorphism, ring automorphism, field automorphism.
automorphism group
The automorphisms of any object or structure can be composed to produce new automorphisms of that object. Since functional composition is associative, and since automorphisms are by their nature invertible, the automorphisms of an object or structure form a group (with the trivial automorphism serving as the identity element). This group is called the automorphism group of that object, usually denoted by Aut(X) for some object X. Automorphism groups are very useful, as they yield information about the symmetries of objects at hand.
binary operation
A binary operation on a set X is a function whose domain is the set of ordered pairs of elements from X and whose range is X.
Examples: addition on the set of integers, function composition.
closure
Topology: The closure of a subset E of a topological space is the smallest closed set containing E. It may also be expressed as the union of E with its accumulation points. If E is closed, then it is equal to its closure.
Algebra: An algebraic closure of a field F is a field G containing F such that every polynomial with coefficients from F has a root in G.
coefficient
See polynomial.
commutative
An operation “ · ” on elements of a set A is commutative if for all elements a, b in A, a · b = b · a.
Cf. associative, distributive property.
commutative property
A property of numbers which states that the operations of addition and multiplication are commutative. Cf. distributive property, associative.
commutative ring
A ring in which the multiplication operation is commutative.
complete lattice
A lattice X is complete if every subset of X has a least upper bound and a greatest lower bound in X.
complete ordered field
A linearly ordered field in which every subset with an upper bound has a least upper bound. Every complete ordered field is isomorphic to the set of real numbers.
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