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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.
For example, in modular arithmetic, for every integer m greater than 1, the congruence modulo m is an equivalence relation on the integers, for which two integers a and b are equivalent—in this case, one says congruent—if m divides ; this is denoted ().
In a Euclidean system, congruence is fundamental; it is the counterpart of equality for numbers. In analytic geometry , congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Euclidean distance between them is equal to ...
Similar figures. In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other.More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection.
If C is an additive category and we require the congruence relation ~ on C to be additive (i.e. if f 1, f 2, g 1 and g 2 are morphisms from X to Y with f 1 ~ f 2 and g 1 ~g 2, then f 1 + g 1 ~ f 2 + g 2), then the quotient category C/~ will also be additive, and the quotient functor C → C/~ will be an additive functor.
Furthermore, every congruence-permutable algebra is congruence-modular, i.e. its lattice of congruences is modular lattice as well; the converse is not true however. After Maltsev's result, other researchers found characterizations based on conditions similar to that found by Maltsev but for other kinds of properties.