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Math 619
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 The monounary direct product problem.
(This is problem 3 of section 5.1 of McKenzie, McNulty, Taylor's book.)
All of the algebras A_{3k+1} satisfy the equation
f ^{3}(x) = x
so we can work in the variety of monounary algebras satisfying this
equation. For every element a in every algebra in this variety
there is a smallest positive r such that
f ^{r}(a) = a and this r
must be 1 or 3. Two algebras in the variety are isomorphic iff
then have the same number of elements of order 1 and of order 3.
Now the algebras A_{3k+1} have exactly one element
of order 1 and conversely if A in the variety is finite
and has exactly one element of order 1 then it is
A_{3k+1} for some k. Now the number of elements
of order 1 in the direct product of two algebras in the variety is
the product of the number in each. Both parts of the problem follow
from this.
