A submonoid of a monoid M is a subset N of M containing the unit element, and such that, if x, y ∈ N then x · y ∈ N. It is then clear that N is itself a monoid, under the binary operation induced by that of M. Equivalently, a submonoid is a subset N such that N = N *, where the superscript * is the Kleene star: the set is closed under composition or concatenation of its elements.

Given an element φ of H *, the orthogonal complement of the kernel of φ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set.

*, a set of observed data points, i. e..

*, is English, which implies that the preferred character set encoding for these newsgroups is ASCII.

*, the set of finite bit strings

*, the set of congruence classes modulo

One can show that if f ( z ) is a holomorphic function on an open set D, then f ( z *) is an antiholomorphic function on D *, where D * is the reflection against the x-axis of D, or in other words, D * is the set of complex conjugates of elements of D. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function.

Given a perfect graph G, Lovász forms a graph G * by replacing each vertex v by a clique of t < sub > v </ sub > vertices, where t < sub > v </ sub > is the number of distinct maximum independent sets in G that contain v. It is possible to correspond each of the distinct maximum independent sets in G with one of the maximum independent sets in G *, in such a way that the chosen maximum independent sets in G * are all disjoint and each vertex of G * appears in a single chosen set ; that is, G * has a coloring in which each color class is a maximum independent set.

* The Derek Trucks * and Susan Tedeschi * Band joined by Warren Haynes, Sheryl Crow, David Hidalgo ^*, Cesar Rosas *, and Johnny Winter * ( This set replaced the originally scheduled Allman Brothers Band.

Charles de Lint *, writer of many modern and urban fairy tales, set his novel The Little Country in the village of Mousehole in Cornwall.

Over Sea, Under Stone and Greenwitch from the series of fantasy novels The Dark Is Rising, by Susan Cooper *, are set in Cornwall.

*, June 20, 1975 -, drum set ) 1996 – 2002

* Alumni of classes préparatoires ( from scientific preparatory classes MP / MP *, but also business school preparatory classes ( " HEC math ") and Humanities and Social Sciences preparatory classes ( Khâgne BL )).

Since the relations R generate an equivalence relation on X *, one can consider elements of M to be the equivalence classes of X < sup >*</ sup > under R. For each class

* A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure.

Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by, is the set.

This set, consisting of the integers congruent to a modulo n, is called the congruence class or residue class or simply residue of the integer a, modulo n. When the modulus n is known from the context, that residue may also be denoted.

The set of all congruence classes modulo n is denoted or ( the alternate notation is not recommended because of the possible confusion with the set of n-adic integers ).

The set H of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal.

The prototypical example of a congruence relation is congruence modulo on the set of integers.

For a congruence on a ring, the equivalence class containing 0 is always a two-sided ideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.

A congruence ~ is determined entirely by the set

There are relations which are both symmetric and antisymmetric ( equality and its subrelations, including, vacuously, the empty relation ), there are relations which are neither symmetric nor antisymmetric ( the relation " divides " on the set ℤ ; the relation " preys on " in biological sciences ), there are relations which are symmetric and not antisymmetric ( congruence modulo n ), and there are relations which are not symmetric but are antisymmetric (" is less than or equal to ").

Note that the ( congruence ) closure of a binary relation depends on the operations in, not just on the carrier set.

The proof of correctness for AKS consists of showing that there exists a suitably small r and suitably small set of integers A such that, if the congruence holds for all such a in A, then n must be prime.

* Juxtaposition stands for multiplication on the set of congruence classes or application of the group operation ( as applicable )

The algorithm attempts to set up a congruence of squares modulo n ( the integer to be factorized ), which often leads to a factorization of n. The algorithm works in two phases: the data collection phase, where it collects information that may lead to a congruence of squares ; and the data processing phase, where it puts all the data it has collected into a matrix and solves it to obtain a congruence of squares.

we let Con ( A ) be the set of all congruence relations on A.

* The greatest element of Con ( A ) is the set AxA, which is the congruence corresponding to the constant homomorphism.

An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III, and from V. 1 is impossible.

The resulting roulette is formed by the locus of the generator subjected to the same set of congruence transformations.

In class theories such as Von Neumann – Bernays – Gödel set theory and Morse – Kelley set theory, there is a possible axiom called the axiom of global choice which is stronger than the axiom of choice for sets because it also applies to proper classes.

In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets.

The main classes of chemotaxis receptors are triggered by formyl peptides-formyl peptide receptors ( FPR ), chemokines-chemokine receptors ( CCR or CXCR ) and leukotrienes-leukotriene receptors ( BLT ); however, induction of a wide set of membrane receptors ( e. g. amino acids, insulin, vasoactive peptides ) also elicit migration of the cell.

The completion of M can be constructed as a set of equivalence classes of Cauchy sequences in M. For any two Cauchy sequences ( x < sub > n </ sub >)< sub > n </ sub > and ( y < sub > n </ sub >)< sub > n </ sub > in M, we may define their distance as

But " having distance 0 " is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of M. The original space is embedded in this space via the identification of an element x of M with the equivalence class of sequences converging to x ( i. e., the equivalence class containing the sequence with constant value x ).

Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a field that has the rational numbers as a subfield.

The set of all possible equivalence classes of X by ~, denoted, is the quotient set of X by ~.

The set of all equivalence classes in given an equivalence relation is denoted as and called the quotient set of by.

One way in which the quotient set resembles division is that if is finite and the equivalence classes are all equinumerous, then the number of equivalence classes in can be calculated by dividing the number of elements in by the number of elements in each equivalence class.

Then the equivalence class of the pair can be identified with the rational number, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.

Therefore, the set of all equivalence classes of forms a partition of: every element of belongs to one and only one equivalence class.

Therefore, every educational program that desires to improve students ' outcomes in political, health and economic behavior should include a Socratically taught set of classes to teach logic and critical thinking.

With the above product, the set of all homotopy classes of loops with base point x < sub > 0 </ sub > forms the fundamental group of X at the point x < sub > 0 </ sub > and is denoted

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

The " Palestrina style " now serves as a basis for college Renaissance counterpoint classes, thanks in large part to the efforts of the 18th century composer and theorist Johann Joseph Fux, who, in a book called Gradus ad Parnassum ( Steps to Parnassus, 1725 ), set about codifying Palestrina's techniques as a pedagogical tool for students of composition.

Isomorphism classes are often defined if the exact identity of the elements of the set is considered irrelevant, and the properties of the structure of the mathematical object are studied.

Traditional MUDs implement a role-playing video game set in a fantasy world populated by fictional races and monsters, with players choosing classes in order to gain specific skills or powers.

In mathematics, model theory is the study of ( classes of ) mathematical structures ( e. g. groups, fields, graphs, universes of set theory ) using tools from mathematical logic.