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
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1.
is a subalgebra of , and
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2.
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:
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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.
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If is the ideal consisting of all finite subsets of , then is called the weak direct product of .
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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 .
References
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
Title | restricted direct product of algebraic systems |
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Canonical name | RestrictedDirectProductOfAlgebraicSystems |
Date of creation | 2013-03-22 17:05:57 |
Last modified on | 2013-03-22 17:05:57 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 7 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 08B25 |
Defines | restricted direct product |
Defines | weak direct product |