For example, let be the direct product of and . Define on as follows:
The equivalence of (1) and (2) follows from the assumption that for each , so that . Similarly define
In fact, we have the following:
Clearly, if , then they are permutable. Conversely, suppose they are permutable. Let and . We want to show that . If , then there is such that and , so . This shows . If , then there is such that and with . If , then we are done, since (as an element belonging to, say , can be written as ). If and then we are done too, since this is just the definition of . If and , then , by permutability. ∎
Remark. From the example above, it is not hard to see that an algebraic system is the direct product of two algebraic systems iff there are two permutable congruences and on such that and , where is the diagonal relation on , and that and . This result can be generalized to arbitrary direct products.
|Date of creation||2013-03-22 17:09:29|
|Last modified on||2013-03-22 17:09:29|
|Last modified by||CWoo (3771)|