Product Monoids and Projection
Let
denote an n-tuple of alphabets . Let denote all possible combinations of finite-length strings from the alphabets:
(In more formal language, is the Cartesian product of the free monoids of the . The superscript star is the Kleene star.) Composition in the product monoid is component-wise, so that, for
and
then
for all in . Define the union alphabet to be
(The union here is the set union, not the disjoint union.) Given any string, we can pick out just the letters in some using the corresponding string projection . A distribution is the mapping that operates on with all of the, separating it into components in each free monoid:
Read more about this topic: History Monoid
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