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group – inf
group
A set together with a binary operation defined on its elements, satisfying - The group operation is associative,
- There is an identity element, i.e., an element e of the group so that for any other element a of the group we have ea = ae = a, and
- for every element a of the group there is an element a´, called the inverse of a, so that aa´ = a´a = e.
If the group operation is commutative, the group is called Abelian. If the condition that every element have an inverse is dropped, the set is called a monoid. If in addition the requirement that an identity element exist is dropped, the set is called a semigroup. Finally, if the condition that the group operation be associative is dropped, the set is called a groupoid.
 Cf. field, module, ring
group automorphism
A group isomorphism from a group to itself. That is, a bijective function from a group to itself that preserves the group operation, i.e. f(ab) = f(a)f(b) for all a, b.
Cf. group homomorphism, group isomorphism.
group homomorphism
A function from one group to another that preserves the group operation, i.e., f(ab) = f(a)f(b) for all a, b.
Cf. group automorphism, group isomorphism.
group isomorphism
A group homomorphism that is both “one-to-one” and “onto.” That is, a bijective function from one group to another that preserves the group operation, i.e. f(ab) = f(a)f(b) for all a, b.
Cf. group homomorphism, group automorphism.
groupoid
See group.
Hempel’s Ravens Paradox
ARTICLE
A paradox of inductive logic described by the philosopher C.G. Hempel. See the article for a complete exposition.
homomorphism
A function f from one algebra to another is called a homomorphism if it preserves operations on elements, that is, if for any a, b in the domain, f(ab) = f(a)f(b).
Cf. group homomorphism, ring homomorphism.
hyperset
A set which is not well-founded, i.e., which involves self-membership or, equivalently, an infinite descending membership chain. Example: the Quine atom x = {x}.
hypotenuse
On a right triangle, the side opposite the right angle.
i
See imaginary number.
icosahedron
A polyhedron having twenty faces.
The faces of a regular icosahedron are congruent, equilateral triangles.
Cf. Platonic solid.
idempotent
An element of an algebra is called idempotent if it is equal to its own square.
If for a binary operation “ · ” on an algebra A we have that a · a = a for every a in A, then “ · ” is called an idempotent operation.
identity element
Given an algebra X with a binary operation “ • ”, an identity element of X is an element e such that for all x in X, x • e = e • x = x. If “ • ” is addition, the identity element is usually called zero and denoted by 0; if the operation is multiplication, the identity element is usually called unity and denoted by 1.
It is possible for an algebra to have distinct left and right identities, that is, distinct e and e’ such that for all x, e • x = x and x • e’ = x. If an element is both a left and right identity, then it is the unique identity element in the algebra for that operation.
identity function
A function that maps each domain element to itself. Also called the identity map.
identity map
See identity function.
image
Given a function f with domain X, the image under f of a subset A of X, denoted f(A), is the subset of the range consisting of those elements to which elements of A are mapped by f.
imaginary number
By definition, the square root of –1, i.e., i 2 = –1.
Cf. complex number.
inaccessible cardinal
A cardinal k is called weakly inaccessible if and only if it is a regular limit cardinal. A cardinal k is called strongly inaccessible if and only if k > w and for every cardinal l less than k, 2l is less than k. If GCH is assumed, then weakly inaccessibles are strongly inaccessible. Ordinary set theory (ZFC) cannot prove that weakly inaccessibles exist.
Cf. König’s Lemma.
incidence matrix
A way to represent a graph as a matrix. Given a graph G with vertices {v1, v2, ..., vn} and edges {e1, e2, ..., em}, the incidence matrix of G is an n by m matrix, whose i, j entry is the number of times ej is incident on vi (zero if not incident, one if incident and not a loop, two if incident and a loop).
Cf. adjacency matrix.
independent axiom
In a formal mathematical theory, a formula or statement in the theory is said to be independent if it cannot be derived (proved, deduced) from the other axioms of the theory.
inf
Abbreviation of infimum.
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