In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector...

integer factorization algorithms Matrix congruence, an equivalence relation between two matrices Congruence (manifolds), in the theory of smooth manifolds...

integer k such that a − b = k m. Congruence modulo m is a congruence relation, meaning that it is an equivalence relation that is compatible with the operations...

structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed....

where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of...

using a congruence relation. Quotient algebras are also called factor algebras. Here, the congruence relation must be an equivalence relation that is...

In number theory, the Eichler–Shimura congruence relation expresses the local L-function of a modular curve at a prime p in terms of the eigenvalues of...

(an element of the Euclidean group E(n)) with f(A) = B. Congruence is an equivalence relation. Two conic sections are congruent if their eccentricities...

whether a homomorphism is injective. In these cases, the kernel is a congruence relation. This article is a survey for some important types of kernels in...

when one may be moved to coincide with the other, and the equality/congruence relation is the isomorphism classes of isometries between shapes. Similarly...

(x{\bmod {n}}_{1},\ldots ,x{\bmod {n}}_{k})} maps congruence classes modulo N to sequences of congruence classes modulo ni. The proof of uniqueness shows...

(known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group, the...

In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations...

\operatorname {Tolr} (A)} under inclusion. Since every congruence relation is a tolerance relation, the congruence lattice Cong ⁡ ( A ) {\displaystyle \operatorname...

In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms. Given a positive integer n, Fermat's factorization...

for every x, y, u, v in S. Like any equivalence relation, a semigroup congruence ~ induces congruence classes [a]~ = {x ∈ S | x ~ a} and the semigroup...

Let C be a category. A congruence relation R on C is given by: for each pair of objects X, Y in C, an equivalence relation RX,Y on Hom(X,Y), such that...

U+225D ≝ EQUAL TO BY DEFINITION or U+2254 ≔ COLON EQUALS), or a congruence relation in modular arithmetic. Also, in chemistry, the triple bar can be...

that ad ≡ 1 (mod p) holds trivially for a ≡ 1 (mod p), because the congruence relation is compatible with exponentiation. And ad = a20d ≡ −1 (mod p) holds...

approximation – Approximation of powers of some binomials Congruence relation – Equivalence relation in algebra Double tilde (disambiguation) – Various meanings...

multiplication defined in the next section. The congruence relation, modulo m, partitions the set of integers into m congruence classes. Operations of addition and...

(m_{2},n_{2})\equiv (m_{1}m_{2},n_{1}n_{2}).} This equivalence relation is a congruence relation, which means that it is compatible with the addition and multiplication...

satisfies the congruence relation: b n ≡ b ( mod n ) {\displaystyle b^{n}\equiv b{\pmod {n}}} for all integers ⁠ b {\displaystyle b} ⁠. The relation may also...

{\displaystyle S} such that the syntactic congruence defined by S {\displaystyle S} is the equality relation. Let us call [ s ] S {\displaystyle [s]_{S}}...

R/I} and called the quotient of R by I. (It is an instance of a congruence relation and is a generalization of modular arithmetic.) If the ideal I is...

For non-negative integers m and n and a prime p, the following congruence relation holds: ( m n ) ≡ ∏ i = 0 k ( m i n i ) ( mod p ) , {\displaystyle...

that the above congruence holds trivially for a ≡ 1 ( mod p ) {\displaystyle a\equiv 1{\pmod {p}}} , because the congruence relation is compatible with...

quotient algebra obtained by factoring the algebra of formulas by this congruence relation. The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski...

(This relation is interpreted inclusively, so that Bxyz is trivially true whenever x=y or y=z). Congruence (or "equidistance"), a tetradic relation. The...

n} , n {\displaystyle n} is prime if and only if the polynomial congruence relation holds within the polynomial ring ( Z / n Z ) [ X ] {\displaystyle...