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Math 619

Homework Solutions

  1. The mono-unary direct product problem. (This is problem 3 of section 5.1 of McKenzie, McNulty, Taylor's book.)

    All of the algebras A3k+1 satisfy the equation f 3(x) = x so we can work in the variety of mono-unary 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 A3k+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 A3k+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.