A semigroup and Gaussian polynomials

Words are defined to be sequences @x: Z">="0 -> Z">"0 @? [~] with @x(n) = ~ for almost all n. I(@x) denotes the number of inversions in @x; i.e., the number of pairs (m, n) with m@x(n). The product, @x * @z, is defined to be a ''meshed juxtaposition'' of @x with the word @z' = @z + maximum integer in @x. (Terms of @z' are meshed into the spaces '~' in @x). With these definitions words form a unique factorization (hence cancellative) semigroup, and I(@x * @z) = I(@x) + I(@z). This is used in conjunction with Netto-MacMahon type generating functions, @?"@?"@e"A@g^I^(^@?^) to give a purely combinatorial treatment of the Gaussian polynomials. Factorization theorems and Chu-Vandermonde and Euler identities are proved.