BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry		field – greatest lower bound field   A set together with two binary operations (called addition and multiplication) defined on its elements, and satisfying the set is an Abelian group under the addition operation, the set is a commutative ring with addition as the group operation and multiplication as the ring operation, and every element except the additive identity has a multiplicative inverse (i.e., the set is a division ring). In other words, a field is a commutative division ring. If in addition the field is linearly ordered, it is called an ordered field. An ordered field is complete if every subset of the field with an upper bound has a least upper bound. The real numbers are fully characterized by the fact that they form a complete ordered field, that is, every complete ordered field is isomorphic to the real numbers. field automorphism   A field isomorphism from a field to itself. That is, a bijective function from a field to itself that preserves addition and multiplication, i.e. f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b). Cf. field isomorphism. field homomorphism   A function from one field to another that preserves addition and multiplication, i.e. f(a+b) = f(a) + f(b) and f(ab) = f(a)f(b). All field homomorphisms are actually isomorphisms of the domain with some subfield of the range. Cf. field isomorphism, field automorphism. field isomorphism   A field homomorphism that is both “one-to-one” and “onto,” that is, a bijective function from one field to another that preserves addition and multiplication, i.e. f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b). Cf. field automorphism. figure   General: A drawing, picture, or illustration, usually accompanying a text description. Also synonymous with digit or numeral. Geometry: A graphical (visual) representation of points, lines, curves, surfaces, solids, or regions. The word “figure” may also be used to refer to the abstract object or set of points thus represented. Cf. plane figure. filter   If X is a set (or class) and F is a family of subsets of X, then F is called a filter provided F is closed under intersections, i.e., for any sets A, B in F their intersection is also in F, and F is closed under supersets (upwardly closed), i.e., if A is any set in F and B is any set in X containing A, then B is in F. If X and F are as above, and if in addition for every subset Y of X either Y or its complement is in F, then F is called an ultrafilter (or maximal filter). Equivalently, an ultrafilter is a filter that is not the proper subset of any filter. A free ultrafilter is an ultrafilter which contains the complement of every finite set. If A is any subset of X, then the collection of supersets of A together with all finite intersections of supersets of A is called the filter generated by A. More generally, if U is any family of subsets of X that is closed unter finite intersections, then the family F of subsets which includes precisely U and all of the intersections of supersets of members of U is the filter generated by U, and U is called its filter base. If a filter F is generated by a singleton set, then it is called a principal filter. finite intersection property   A collection of sets has the finite intersection property if every finite subcollection has non-empty intersection. finite set   A set X is finite if there is a natural number n (possibly 0) such that X can be said to have exactly n elements. More formally, a set is finite if it is not bijective with any proper subset of itself. flat pair   A pair of elements {a,b} with no order defined on the elements. Sometimes also called an unordered pair. Cf. ordered pair. formal polynomial   See polynomial. function   Given a binary relation R on sets A and B, we say that the R is a function if each element of A is paired with at most one element of B. In this case we call A the domain set, and we call B the range set. If f is such a function and the ordered pair (x,y) is an element of the function, we typically write f(x) = y. More generally, if R is an n-place relation, then R is a function of n-1 variables if each (n-1)-tuple is matched with at most one element of the range set (i.e., the nth set of the Cartesian product on which the relation is defined). If in addition there is at most one element of the domain set matched with any given element of the range set, then the function is called “one-to-one” or injective. If the function maps at least one element of the domain set to every element of its range set, then it is called “onto” or surjective. A function which is both injective and surjective is called bijective. Functions with range in the real numbers or complex numbers are called real-valued or complex-valued respectively. functional calculus   See predicate calculus. Fundamental Theorem of Algebra   The field of complex numbers is algebraically closed, that is, any polynomial with complex coefficients has a complex root. Fundamental Theorem of Arithmetic    ARTICLE   Every natural number is either prime or may be decomposed uniquely into prime factors. g.c.d.   Abbreviation for greatest common divisor. GCH   See: generalized continuum hypothesis. generalized continuum hypothesis   The claim that there is no set of intermediate cardinality between any given set and its power set. The GCH is independent of (i.e., cannot be proved true or false by) the known axioms of set theory. Cf. continuum hypothesis. Related article: Gödel's Theorems Related MiniText: Infinity -- You Can't Get There From Here... g.l.b.   Abbreviation for greatest lower bound. graph   General: A picture or diagram representing one or more relationships among certain objects or quantities. Functions: The collection of ordered tuples (x,y), consisting of domain elements x (possibly also tuples) and corresponding range elements y. Although informally we often mean the picture when we speak of the graph of a function, the graph should always be formally thought of as a set of tuples. Graph Theory: A collection of nodes (also called vertices), together with a set of edges joining the nodes. Formally, a graph G is a set of vertices V, a set of edges E, and a symmetric relation between them specifying incidence. Every edge must be incident on exactly two (not necessarily distinct) vertices, called its ends. Two vertices incident on the same edge are called adjacent. Graphs are often represented pictorially, with the vertices being points (or small circles or squares or some such), and the edges being curves (often line segments), with the endpoints of each such curve being the ends of that edge in the graph. Cf. Konigsberg Bridge Problem. greatest common divisor   The greatest common divisor of two natural numbers a and b is the largest natural number c such that c divides a and c divides b. greatest lower bound   A lower bound which is greater than or equal to every other lower bound. Cf. Least upper bound.
		field – greatest lower bound

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