reduced direct product
defined above is a congruence relation on .
Since is an ideal, , so , and is an equivalence relation on .
Next, let be an -ary operator on and , where . We want to show that . Let be the associated -ary operators on . If , then , which implies that for some . This implies that
Since is an ideal, and each , we have that as well, this means that . ∎
Definition. Let , be a Boolean ideal of and be defined as above. The quotient algebra is called the -reduced direct product of . The -reduced direct product of is denoted by . Given any element , its image in the reduced direct product is given by , or for short.
Example. Let , and let be the principal ideal generated by . Then . The congruence is given by iff or . This implies that for all . In other words, is isomorphic to the direct product of . Therefore, the -reduced direct product of is isomorphic to .
The example above can be generalized: if , then
For , write . It is not hard to see that the map given by is the required isomorphism.
Remark. The definition of a reduced direct product in terms of a Boolean ideal can be equivalently stated in terms of a Boolean filter . All there is to do is to replace by its complement: . The congruence relation is now , where is the ideal complement of . When is prime, the -reduced direct product is called a prime product, or an ultraproduct, since any prime filter is also called an ultrafilter. Ultraproducts can be more generally defined over arbitrary structures.
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
|Title||reduced direct product|
|Date of creation||2013-03-22 17:10:11|
|Last modified on||2013-03-22 17:10:11|
|Last modified by||CWoo (3771)|