restricted direct product of algebraic systems
Let be a family of algebraic systems indexed by a set . Let be a Boolean ideal in , the Boolean algebra over the power set of . A subset of the direct product is called a restricted direct product of if
is a subalgebra of , and
given any , we have that iff .
If it is necessary to distinguish the different restricted direct products of , we often specify the “restriction”, hence we say that is a -restricted direct product of , or that is restricted to .
Here are some special restricted direct products:
If above, then is the direct product , for if , then clearly , where ( is non-empty since it is a subalgebra). Therefore .
This justifies calling the direct product the “unrestricted direct product” by some people.
If is the ideal consisting of all finite subsets of , then is called the weak direct product of .
If is the singleton , then is also a singleton: pick , then , which is equivalent to saying that .
Remark. While the direct product of always exists, restricted direct products may not. For example, in the last case above, A -restricted direct product exists only when there is an element that is fixed by all operations on it: that is, if is an -ary operation on , then . In this case, is a -restricted direct product of .
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
|Title||restricted direct product of algebraic systems|
|Date of creation||2013-03-22 17:05:57|
|Last modified on||2013-03-22 17:05:57|
|Last modified by||CWoo (3771)|
|Defines||restricted direct product|
|Defines||weak direct product|