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If $S$ is a finite subset of a commutative monoid, one can use the notation

$\prod_{x \in S} {x}$

for denoting the product of all elements of $S$.

It is rather obvious, that the same notation can also be used if $S$ is a finite commutative subset of a non-commutative monoid.

I am looking for a reference in the literature (algebra textbooks), where the latter case ($S$ is a finite commutative subset of a non-commutative monoid) was considered.

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Many examples can be probably easily be found in literature using differential operators, e.g. when discussing hypergeometric differential operators one will encounter products over integers (factorials) and (linear) polynomials (upper/lower factorial sequences), both of which lie in the center of the Weyl algebra $\rm\,\mathbb C\big\langle x,\frac{d}{dx}\big\rangle.\ \ $ –  Bill Dubuque Jul 15 '12 at 19:10
    
Why do you need a reference? Just restrict yourself to working in the commutative monoid generated by $S$. –  Qiaochu Yuan Jul 16 '12 at 1:00

1 Answer 1

up vote 1 down vote accepted

Theorem 18.7 in "Modern Algebra" by Seth Warner, p.151. (Dover Publications, New York, 1990).

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