congruence relation on an algebraic system
For example, and are both congruence relations on . is called the trivial congruence (on ). A proper congruence relation is one not equal to .
The set of congruences of an algebraic system is a complete lattice. The meet is the usual set intersection. The join (of an arbitrary number of congruences) is the join of the underlying equivalence relations (http://planetmath.org/PartitionsFormALattice). This join corresponds to the subalgebra (of ) generated by the union of the underlying sets of the congruences. The lattice of congruences on is denoted by .
(restriction) If is a congruence on and is a subalgebra of , then defined by is a congruence on . The equivalence of is obvious. For any -ary operator inherited from ’s , if , then . Since both and are in , as well. is the congruence restricted to .
(extension) Again, let be a congruence on and a subalgebra of . Define by . In other words, iff for some . We assert that is a subalgebra of . If is an -ary operator on and , then , so . Since , . Therefore, is a subalgebra. Because , we call it the extension of by .
Let be a subset of . The smallest congruence on such that for all is called the congruence generated by . is often written . When is a singleton , then we call a principal congruence, and denote it by .
Furthermore, for each -ary operator on , define by
It is easy to see that is a well-defined operator on . The -algebra thus constructed is called the quotient algebra of over .
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
|Title||congruence relation on an algebraic system|
|Date of creation||2013-03-22 16:26:23|
|Last modified on||2013-03-22 16:26:23|
|Last modified by||CWoo (3771)|
|Defines||congruence restricted to a subalgebra|
|Defines||extension of a subalgebra by a congruence|
|Defines||congruence generated by|