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In abstract algebra, a **congruence relation** (or simply **congruence**) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements.^{ [1] } Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or **congruence classes**) for the relation.^{ [2] }

The prototypical example of a congruence relation is congruence modulo on the set of integers. For a given positive integer , two integers and are called **congruent modulo **, written

if is divisible by (or equivalently if and have the same remainder when divided by ).

For example, and are congruent modulo ,

since is a multiple of 10, or equivalently since both and have a remainder of when divided by .

Congruence modulo (for a fixed ) is compatible with both addition and multiplication on the integers. That is,

if

- and

then

- and

The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo is a congruence relation on the ring of integers, and arithmetic modulo occurs on the corresponding quotient ring.

The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions of congruence can be made for groups, rings, vector spaces, modules, semigroups, lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operations are well-defined on the equivalence classes.

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If is a group with operation , a **congruence relation** on is an equivalence relation on the elements of satisfying

- and

for all . For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the cosets of this subgroup. Together, these equivalence classes are the elements of a quotient group.

When an algebraic structure includes more than one operation, congruence relations are required to be compatible with each operation. For example, a ring possesses both addition and multiplication, and a congruence relation on a ring must satisfy

- and

whenever and . 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.

The general notion of a congruence relation can be formally defined in the context of universal algebra, a field which studies ideas common to all algebraic structures. In this setting, a relation on a given algebraic structure is called **compatible** if

- for each and each -ary operation defined on the structure: whenever and ... and , then .

A congruence relation on the structure is then defined as an equivalence relation that is also compatible.^{ [3] }^{ [4] }

If is a homomorphism between two algebraic structures (such as homomorphism of groups, or a linear map between vector spaces), then the relation defined by

- if and only if

is a congruence relation on . By the first isomorphism theorem, the image of *A* under is a substructure of *B* isomorphic to the quotient of *A* by this congruence.

On the other hand, the relation induces a unique homomorphism given by

- .

Thus, there is a natural correspondence between the congruences and the homomorphisms of any given structure.

In the particular case of groups, congruence relations can be described in elementary terms as follows: If *G* is a group (with identity element *e* and operation *) and ~ is a binary relation on *G*, then ~ is a congruence whenever:

- Given any element
*a*of*G*,*a*~*a*(**reflexivity**); - Given any elements
*a*and*b*of*G*, if*a*~*b*, then*b*~*a*(**symmetry**); - Given any elements
*a*,*b*, and*c*of*G*, if*a*~*b*and*b*~*c*, then*a*~*c*(**transitivity**); - Given any elements
*a*,*a'*,*b*, and*b'*of*G*, if*a*~*a'*and*b*~*b'*, then*a***b*~*a'***b'*; - Given any elements
*a*and*a'*of*G*, if*a*~*a'*, then*a*^{−1}~*a'*^{−1}(this can actually be proven from the other four, so is strictly redundant).

Conditions 1, 2, and 3 say that ~ is an equivalence relation.

A congruence ~ is determined entirely by the set {*a* ∈ *G* : *a* ~ *e*} of those elements of *G* that are congruent to the identity element, and this set is a normal subgroup. Specifically, *a* ~ *b* if and only if *b*^{−1} * *a* ~ *e*. So instead of talking about congruences on groups, people usually speak in terms of normal subgroups of them; in fact, every congruence corresponds uniquely to some normal subgroup of *G*.

A similar trick allows one to speak of kernels in ring theory as ideals instead of congruence relations, and in module theory as submodules instead of congruence relations.

A more general situation where this trick is possible is with Omega-groups (in the general sense allowing operators with multiple arity). But this cannot be done with, for example, monoids, so the study of congruence relations plays a more central role in monoid theory.

The general notion of a congruence is particularly useful in universal algebra. An equivalent formulation in this context is the following:^{ [4] }

A congruence relation on an algebra *A* is a subset of the direct product *A*×*A* that is both an equivalence relation on *A* and a subalgebra of *A*×*A*.

The kernel of a homomorphism is always a congruence. Indeed, every congruence arises as a kernel. For a given congruence ~ on *A*, the set *A*/~ of equivalence classes can be given the structure of an algebra in a natural fashion, the quotient algebra. The function that maps every element of *A* to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.

The lattice **Con**(*A*) of all congruence relations on an algebra *A* is algebraic.

John M. Howie described how semigroup theory illustrates congruence relations in universal algebra:

- In a group a congruence is determined if we know a single congruence class, in particular if we know the normal subgroup which is the class containing the identity. Similarly, in a ring a congruence is determined if we know the ideal which is the congruence class containing the zero. In semigroups there is no such fortunate occurrence, and we are therefore faced with the necessity of studying congruences as such. More than anything else, it is this necessity that gives semigroup theory its characteristic flavour. Semigroups are in fact the first and simplest type of algebra to which the methods of universal algebra must be applied…
^{ [5] }

- ↑ Hungerford, Thomas W..
*Algebra*. Springer-Verlag, 1974, p. 27 - ↑ Hungerford, 1974, p. 26
- ↑ Henk Barendregt (1990). "Functional Programming and Lambda Calculus". In Jan van Leeuwen (ed.).
*Formal Models and Semantics*. Handbook of Theoretical Computer Science.**B**. Elsevier. pp. 321–364. ISBN 0-444-88074-7. Here: Def.3.1.1, p.338. - 1 2 Clifford Bergman, Universal Algebra: Fundamentals and Selected Topics, Taylor & Francis (2011), Sect. 1.5 and Exercise 1(a) in Exercise Set 1.26 (Bergman uses the expression
*having the substitution property*for*being compatible*) - ↑ J. M. Howie (1975)
*An Introduction to Semigroup Theory*, page v, Academic Press

In number theory, the **Chinese remainder theorem** states that if one knows the remainders of the Euclidean division of an integer *n* by several integers, then one can determine uniquely the remainder of the division of *n* by the product of these integers, under the condition that the divisors are pairwise coprime.

In mathematics, an **equivalence relation** is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation.

In mathematics, when the elements of some set have a notion of equivalence defined on them, then one may naturally split the set into **equivalence classes**. These equivalence classes are constructed so that elements and belong to the same **equivalence class** if, and only if, they are equivalent.

A **quotient group** or **factor group** is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. For example, the cyclic group of addition modulo *n* can be obtained from the group of integers under addition by identifying elements that differ by a multiple of *n* and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.

In algebra, a **homomorphism** is a structure-preserving map between two algebraic structures of the same type. The word *homomorphism* comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German *ähnlich* meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In mathematics, **modular arithmetic** is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the **modulus**. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book *Disquisitiones Arithmeticae*, published in 1801.

In mathematics, a **semigroup** is an algebraic structure consisting of a set together with an associative binary operation.

In group theory, a branch of mathematics, given a group *G* under a binary operation ∗, a subset *H* of *G* is called a **subgroup** of *G* if *H* also forms a group under the operation ∗. More precisely, *H* is a subgroup of *G* if the restriction of ∗ to *H* × *H* is a group operation on *H*. This is usually denoted *H* ≤ *G*, read as "*H* is a subgroup of *G*".

In algebra, the **kernel** of a homomorphism is generally the inverse image of 0. An important special case is the kernel of a linear map. The kernel of a matrix, also called the *null space*, is the kernel of the linear map defined by the matrix.

In mathematics, specifically abstract algebra, the **isomorphism theorems** are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

In number theory, a **Gaussian integer** is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as **Z**[*i*]. This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.

In ring theory, a branch of abstract algebra, a **quotient ring**, also known as **factor ring**, **difference ring** or **residue class ring**, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring *R* and a two-sided ideal *I* in *R*, a new ring, the quotient ring *R* / *I*, is constructed, whose elements are the cosets of *I* in *R* subject to special *+* and *⋅* operations.

In mathematics, a **quotient algebra** is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are also called **factor algebras**. Here, the congruence relation must be an equivalence relation that is additionally *compatible* with all the operations of the algebra, in the formal sense described below. Its equivalence classes partition the elements of the given algebraic structure. The quotient algebra has these classes as its elements, and the compatibility conditions are used to give the classes an algebraic structure.

In number theory, given a positive integer *n* and an integer *a* coprime to *n*, the **multiplicative order** of *a* modulo *n* is the smallest positive integer *k* such that .

In mathematics, a **congruence subgroup** of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are *even*. More generally, the notion of **congruence subgroup** can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.

In number theory and algebraic geometry, a **modular curve***Y*(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane **H** by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, **Z**). The term modular curve can also be used to refer to the **compactified modular curves***X*(Γ) which are compactifications obtained by adding finitely many points to this quotient. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field of rational numbers **Q** or a cyclotomic field **Q**(ζ_{n}). The latter fact and its generalizations are of fundamental importance in number theory.

In mathematics and computer science, the **syntactic monoid** of a formal language is the smallest monoid that recognizes the language .

In mathematics, **Wolstenholme's theorem** states that for a prime number , the congruence

In mathematics, particularly in the area of number theory, a **modular multiplicative inverse** of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as

In mathematics, a **canonical map**, also called a **natural map**, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. In general, it is the map which preserves the widest amount of structure, and it tends to be unique. In the rare cases where latitude in choices remains, the map is either conventionally agreed upon to be the most useful for further analysis, or sometimes the most elegant map known up to date.

- Horn and Johnson,
*Matrix Analysis,*Cambridge University Press, 1985. ISBN 0-521-38632-2. (Section 4.5 discusses congruency of matrices.) - Rosen, Kenneth H (2012).
*Discrete Mathematics and Its Applications*. McGraw-Hill Education. ISBN 978-0077418939.

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