General Algebra and Linear Algebra

Discussion in 'Math Research' started by Tim Boykett, Aug 11, 2005.

  1. Tim Boykett

    Tim Boykett Guest

    Dear All,

    I have come across an interesting property of some special
    equations on algebras that seem to connect to congruence relations. The
    following is an attempt to formulate the idea of the
    equations in a more general form. I would be most interested in
    any small pointers you might have to the literature. Please excuse any
    misuses of language!

    Let t_i be a set of terms on the variables x_j in
    an algebra with set A.
    Let a_i be some real numbers.
    Let v map A to the reals R such that the equation
    sum_i a_i v(t_i) = 0
    is satisfied for all values of the x_j selected from A. It is clear that
    the set of v satisfying such equations forms a vector space that is
    a subspace of R^A.

    It looks like it should be called something like "solutions of
    systems of equations over valuations of an algebra"
    but of course "valuations" already has a well defined meaning.
    Perhaps it has some more meaning when we generalise R to allow
    it to be an abelian group and the a_i to be integers: then an
    equation like
    v(x*y) - v(x) - v(y) = 0
    corresponds to v being a homomorphism of A into the group R.

    I hope for some clever suggestions,


    Tim Boykett, Aug 11, 2005
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