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I want to introduce formal linear combinations in an upper-level undergraduate combinatorics class. By this I mean expressions like $7 \operatorname{cat} + 5 \operatorname{dog} - \sqrt{2} \operatorname{molerat}$ (and infinite versions thereof), as well as (situation-dependent) rules of multiplication on such expressions.

Formally speaking, this means that I have a commutative ring $R$ and define the free $R$-module $R^{(I)}$ for a set $I$ (usually not $\left\{1,2,\ldots,n\right\}$) and its "completion" $R^I$, and define some $R$-algebra structures on these modules. I will briefly introduce commutative rings first, but I cannot expect anyone to have prior experience with them, and so I would like to avoid direct sum, direct products, universal properties, etc.. The two examples I really want to get to are the following:

  • The polynomial ring $R\left[X\right]$ and the ring $R\left[\left[X\right]\right]$ of formal power series. (A major pain point with US student audiences is having to unteach the misconceptions that polynomials are polynomial functions and that power series have to converge.)

  • The group ring $R\left[S_n\right]$ of the symmetric group. (The students have seen a lot of permutations; my plan is to let them have some fun with sums of permutations -- e.g., Young-Jucys-Murphy elements, Tsetlin libraries, etc.)

But I am interested in other, motivating examples -- ideally baby versions of these objects I can have them play around with that show the structure and give some hint of its usefulness (which means, in particular, that there should be a multiplication -- linear combinations per se are boring). I know of only one for now:

  • The Hardy-Weinberg principle can be proven using a tailor-made non-associative ring: Consider the free module with basis $\left(a,A\right)$, and define a (non-associative) multiplication on it by $a^2 = a$ and $A^2 = A$ and $aA = Aa = \dfrac{1}{2}\left(a+A\right)$. This multiplication is literal multiplication, as in "be fruitful and multiply", as $xy$ is the (probability distribution of) the allele of the offspring of an individual with allele $x$ with an individual with allele $y$. Now, the Hardy-Weinberg principle boils down to the identity $\left(pa+qA\right)^2 = \left(p+q\right)\left(pa+qA\right)$ in this ring, which is easily checked.

I suspect this is just an example for more general probability-type implications -- is there such a thing as a Markov chain with several transition matrices?

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