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...uence]] are Goursat congruences, the complete sequence consists of Goursat congruences. Named after E. Goursat, who studied congruences of this type.

The fully-characteristic congruences of an algebraic system $ A $ of all congruences of $ A $.

...al surface of it (see [[#References|[1]]]). With every Laplace sequence of congruences there is associated a Laplace sequence of focal surfaces (see [[#References

...re $a_i, b_i \in A$, $i=1,\ldots,n$, and $\omega$ is an $n$-ary operation. Congruences in algebraic systems are defined in a similar way. Thus, the equivalence cl ...lattice of relations is not a congruence. The product $\pi_1\pi_2$ of two congruences $\pi_1$ and $\pi_2$ is a congruence if and only if $\pi_1$ and $\pi_2$ comm

...l surfaces of a Guichard congruence are called Guichard surfaces. Guichard congruences are named after G. Guichard (1889), who was the first to consider them. ...top"> G. Guichard, "Surfaces rapporteés à leurs lignes asymptotiques et congruences rapporteés à leurs développables" ''Ann. Sc. Ec. Norm. Sup. (3)'' , '''

...face formed by a conjugate net of lines the tangents to which form two $W$-congruences — a so-called singular conjugate system. Only the Demoulin surfaces permi ...rojective deformation is related to $R$-congruences, which are special $W$-congruences (see [[#References|[a1]]] and [[#References|[2]]], [[#References|[3]]]).

Examples of congruences: a $W$-congruence, in which the asymptotic lines on the focal surfaces corr ...Congruences of arbitrary lines (curves) in a space are called curvilinear congruences.

...rrespond to each other by the orthogonality of the line elements, then the congruences formed by the rays passing through the points on $ S $ Such congruences were examined for the first time by A. Ribaucour in 1881.

===Definition of $\Sigma$-congruences=== For a $\Sigma$-algebra $A$, the set $C(A)$ of all $\Sigma$-congruences for $A$ forms a complete [[Lattice|lattice]] w.r.t. set-theoretic inclusion

...e correspondence with its normal subgroups, and the quotient groups by the congruences are the same as those by the normal subgroups. A quotient group is a normal

c) there exists a family of congruences $ \rho _ {i} $, such that the intersection of these congruences is the identity congruence and $ B/ \rho _ {i} \simeq A _ {i} $

...on series of a universal algebra is defined in terms of congruences. Since congruences in groups are defined by normal subgroups, a composition series of a group ...\Omega$-algebra with subalgebra $E$ such that on any subalgebra of $A$ all congruences commute, then any two normal chains from $E$ to $A$ have isomorphic refinem

...gebras]] with a non-trivial member contains also a member whose lattice of congruences is $2$-element. Such universal algebras are called congruence-simple or sim

...h the modulus is a prime number. A distinguishing feature of the theory of congruences modulo a prime number is the fact that the residue classes modulo $ p $ elements. Congruences modulo a prime number can therefore be treated as equations over finite pri

...ed. A commutative semi-group satisfies the minimum (maximum) condition for congruences if and only if it has a principal series and satisfies the minimum conditio ...I.B. Kozhukov, "On semigroups with minimal or maximal condition on left congruences" ''Semigroup Forum'' , '''21''' : 4 (1980) pp. 337–350</TD></TR><TR><

are different prime numbers, is equivalent to the solvability of the congruences is the number of solutions of (2). Thus, when studying congruences of the form (1) it is sufficient to confine oneself to moduli that are powe

...modulus can be added, subtracted and multiplied in the same way as normal congruences. These operations induce similar operations on the residue classes modulo a then for congruences modulo a double modulus, the analogue of the [[Fermat little theorem|Fermat

...family $(q_\lambda)_{\lambda\in\Lambda}$ is called a separating family of congruences if the intersection of all the $q_\lambda$ is the diagonal congruence (the

* N. Rama Rao, "Some congruences modulo $m$" ''Bull. Calcutta math. Soc.'' '''29''' (1938) 167-170 {{ZBL|64.

...ical class. If in the definition of a radical the discussion is limited to congruences defined by ideals, then another concept of a radical arises, where the corr the intersection of all congruences $ \theta $

...ate cases, may be reduced to the problem of the number of solutions of the congruences $ F ( x _ {1} \dots x _ {n} ) \equiv 0 $( in one variable are the two-term congruences (cf. [[Two-term congruence|Two-term congruence]])

...in projective differential geometry are: the (projective) theory of linear congruences (cf. [[Congruence in geometry|Congruence in geometry]]), and problems on [[ ...ack to the end of the 19th century; the work of G. Darboux on surfaces and congruences was especially important. The first book in which classical projective diff

...on an algebra $\mathbf{A}$ which is expressible as the intersection of all congruences on $\mathbf{A}$ whose factor algebras belong to some fixed variety of $\Ome

...cobson objects, i.e., objects whose prime congruences are meets of maximal congruences. Jacobson categories are axiomatically defined as categories $ \mathbf A

Bianchi congruences were studied by L. Bianchi [[#References|[1]]].

...edekind lattice is a modular lattice. If a universal algebra has commuting congruences, then its congruence lattice is a complete Dedekind lattice [[#References|[

6) for any finite sequence of co-disjunctable congruences $ r _ {1} \dots r _ {n} $ denotes union of congruences on $ A $

Congruences modulo one and the same number can be added, subtracted and multiplied in t The operations of addition, subtraction and multiplication of congruences induce similar operations on the residue classes. Thus, if $ a $

...en Gödelization and coordinatization can be extended to [[Sigma-Congruence|congruences]] defined on $A$ and $C$. A congruence $\sim^\beta \subseteq A \times A$ i

...References|[a5]]]) to a graphical representation of the plactic, or Knuth, congruences. The plactic (or Knuth) congruences are the following: Let the alphabet $X$ be totally ordered and suppose that

...$p(n)$ (see [[#References|[a1]]]) and Ramanujan discovered the surprising congruences $p(5m+4) \equiv 0 \pmod 5$, $p(7m+5) \equiv 0 \pmod 7$, $p(11m+6) \equiv 0

...haracterized by all the conics in the congruence belonging to one quadric. Congruences of conics whose planes form a one-parameter family have one quadruplicate f are the most general one-parameter families, congruences and complexes of conics in $ P _ {3} $.

A [[Semi-group|semi-group]] not containing proper ideals or congruences of some fixed type. Various kinds of simple semi-groups arise, depending on ...ne of which is the null class; and congruence-free semi-groups, not having congruences other than the universal relation and the equality relation.

...ible to calculate the Legendre symbol easily, without resorting to solving congruences. For example,

...ar the ideals, and also on certain relations on semi-groups, in particular congruences. Such restrictions give rise, for example, to various types of simple semi- ...n completely $0$-simple semi-groups, and the far-reaching investigation of congruences on inverse semi-groups; the theory of radicals of semi-groups (cf. [[Radica

congruence if all its classes are convex subsets; the kernel congruences of $ o $- congruences. The decomposition of a totally ordered semi-group $ S $

The conformal-differential geometry in the plane studies sequences and congruences of circles. To a sequence of circles corresponds a curve in three-dimension ...riants of the associated surface in hyperbolic space, and special types of congruences are singled out.

...he semi-group, in particular, the behaviour of $E$ with respect to certain congruences. There exists on any inverse semi-group $S$ a least congruence $\sigma$ wit

The validity of the following congruences has been demonstrated: Certain congruences have also been proved , [[#References|[13]]] for odd $ n $.

and the lattice of all congruences $ \mathop{\rm Con} A $. ...versal algebras with a distributive congruence lattice and with permutable congruences (arithmetic universal algebras) admit a representation as global sections o

...an [[algebraic lattice]], containing the lattice $\mathrm{Con}(A)$ of all congruences on $A$ as a subset (but not necessarily as a sublattice). For properties of

...ed by the set of all subalgebras in a universal algebra, by the set of all congruences in a universal algebra, and by the set of all closed subsets in a topologic

..."Completely 0-simple semigroups: an abstract treatment of the lattice of congruences" , Benjamin (1969)</TD></TR></table>

...td></tr><tr><td valign="top">[a17]</td> <td valign="top"> N. Katz, "The congruences of Clausen–von Staudt and Kummer for Bernoulli–Hurwitz numbers" ''Math

is anti-isomorphic to the lattice of all fully-characteristic congruences (cf. [[Fully-characteristic congruence|Fully-characteristic congruence]]) o is a polarized variety of algebras and the congruences in all algebras in $ \mathfrak M $

...n="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Une interpretation des congruences relatives à la function $\tau$ de Ramanujan" ''Sém. Delange–Pisot–Po

...be imbedded in a complete semi-group, in a simple semi-group (relative to congruences), or in a bi-simple semi-group with zero and an identity (see [[Simple semi

...arise in branches of number theory such as the theory of divisibility, of congruences, of arithmetic functions, of indefinite equations, of partitions, of additi ...ned out to be convenient and instrumental in the development of the theory congruences (cf. [[Congruence|Congruence]]).

satisfies the congruences

...a modular sublattice with a zero and a unit element in the lattice of all congruences on $ S $(

...me congruences are not, in general, Heyting subalgebras or Heyting algebra congruences (nor are the free frames mentioned above free as Heyting algebras). For fre

Lagrange's theorem on congruences: The number of solutions of the [[Congruence|congruence]]

Congruences on a multi-operator group are described by coset classes relative to ideals

the system of congruences $ x \equiv x _ {i} $(

...ebras, then every finitely-generated algebra in the variety is finite. The congruences of any algebra in a variety of universal algebras $M$ of signature $\Omega$ ...variety of all right modules over some associative ring if and only if the congruences on any algebra in $M$ commute, if finite free products (cf. [[Free product|

...on Landweber's exact functor theorem, and on the other hand on interesting congruences for [[Legendre polynomials|Legendre polynomials]]. The general form stated

...bras and for universal algebras with a singleton subalgebra and permutable congruences (see [[#References|[6]]]–[[#References|[8]]]). 3) Various ways of general

...sum of two squares, and asserted one of the basic results of the theory of congruences: $ a ^ {p} - a $ ...eated the basic methods in and completed the construction of the theory of congruences (cf. [[Congruence|Congruence]]), proved the [[Quadratic reciprocity law|qua

...homomorphisms of a quasi-group onto a quasi-group are the so-called normal congruences. (A congruence (cf. [[Congruence (in algebra)|Congruence (in algebra)]]) $ In groups all congruences are normal. A sub-quasi-group $ H $

A new stage in the study of two-term congruences and their applications to other theoretical problems was initiated by I.M.

Many problems in number theory (the theory of congruences, Diophantine

...$p(n)$ (see [[#References|[a1]]]) and Ramanujan discovered the surprising congruences $p(5n+4) \equiv 0 \pmod 5$, $p(7m+5) \equiv 0 \pmod 7$, $p(11m+6) \equiv 0

<TD valign="top"> A.G. Postnikov, "Ergodic problems in the theory of congruences and of Diophantine approximations" , Amer. Math. Soc. (1967) (Translated

...the other via elementary reductions, elementary expansions and unimodular congruences (i.e. transformations $ A \rightarrow P^t A P $,

combines two different problems: To find solutions of congruences (or points of the variety over a finite field) and to find integral or rati

==Subdirect products and congruences.== The congruences form a [[Brouwer lattice|Brouwer lattice]], with the pseudo-complement $

i.e. the system of least non-negative integers satisfying the congruences

...he group of automorphisms, the semi-group of endomorphisms, the lattice of congruences, etc.). The problems involved in the study of the connections between algeb

is subdirectly irreducible if and only if the lattice of congruences of $ L $

is isomorphic to an automorphism group of the lattice of all congruences of the system $ \mathbf A $[[#References|[1]]]. In particular, any inner

Suszko congruences can be defined for the theories of any deductive system, and consequently o Early work on the Suszko and the closely related Leibniz and Tarski congruences discussed below can be found in {{Cite|Ło}}, {{Cite|Sm}}. {{Cite|Su2}} con

...mbination of periods. Hence it follows that (1) is essentially a system of congruences modulo the periods of the differentials $ \phi _ {1} \dots \phi _ {p} $.

..., other geometrical objects in equi-affine space are also studied, such as congruences and complexes of straight lines, vector fields, etc.

and congruences, such as the Ramanujan congruence $ \tau ( n) \equiv \sum _ {d \mid n }

Gauss' reciprocity law has been generalized to congruences of the form

...v_i} \in G_{k_{v_i}}$ and positive integers $m_i$, when does the system of congruences

(For the solvability of all congruences (2) it is sufficient that (2) be solvable for $ g = g _ {0} = 8 D ( q) D in terms of the number of solutions of certain congruences. In case the genus of $ q $

...D> <TD valign="top"> A.G. Postnikov, "Ergodic problems in the theory of congruences and of Diophantine approximations" , Amer. Math. Soc. (1967) (Translated

...ntity relation is compatible with any $\mathcal{F}$, the set of compatible congruences (or equivalences) is not empty for any matrix. Then, it can be proven [2] t Suszko and Leibniz congruences give rise to the $\mathcal{S}$-matrices $\left<\mathfrak{F}_\mathcal{S}/\wi

...iven lattice is isomorphic to the lattice of subsystems, to the lattice of congruences or to some other lattice associated with this algebraic system or universal

...one to embed the Combesqure and Levy transformations of conjugate nets and congruences into classical [[Differential geometry|differential geometry]] [[#Reference

of arithmetical congruences as approximations of periodic sets $ \{ {k _{1} + n \cdot k _ 2} : {n \in

...hed its validity for all $p<100$. In case 1 he showed that (1) implies the congruences ...of the Fermat theorem. They are connected with the solvability of certain congruences or with the existence of prime numbers of a certain form. The equation $x^{

...h.org/legacyimages/d/d032/d032600/d032600205.png" /> may be interpreted as congruences by a "high" degree of <img align="absmiddle" border="0" src="https://www.en

...s of the equation in terms of the number of solutions of the corresponding congruences. The method of trigonometric sums depends less than do other methods on the

...Alg} _ { \vdash } ( \mathcal{L} )$ having equationally definable principal congruences. The Beth definability property for $\mathcal{L}$ is equivalent with surjec

...Euler, in his study of residues remaining in power division, actually used congruences (cf. [[Congruence|Congruence]]) and their division into residue classes, wh

...gebraic geometry over "non-classical" fields originated in the theory of congruences, interpreted as equations over a finite field. It was claimed by Poincaré

...applications of Weil's conjectures in number theory deal with the study of congruences. Already in the case of curves, Weil's theorem entails the best estimate of