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What I mean are expressions like $$ \underbrace{a\cdot a\cdot\ldots \cdot a}_{n}. $$ When $(\cdot)$ is multiplication this obviously is the $n$th power of $a$, when it's addition it's an $n$-fold sum or just "$a$ times $n$". When dealing with a group, ring or field, one would usually call the operations as one of these, but what if I want to stay more general? $n$-fold application does not really fit here. Chain would be one expression that comes to my mind, but that's normally just used in algebraic topology for something rather more complicated.

The thing is, and that's why I'm not quite happy with "power": I actually mean to do this on a field, so there is both a $(\cdot)$ and a $(+)$ around. I want to describe such a generic power, which may be either of $a\cdot\ldots\cdot a$ or $a+\ldots+a$, identified through just a triple $(a,(\cdot),n)$ or $(a,(+),n)$. In the latter case, "power" would certainly be a little confusing, but right: I guess what matters is that it is well defined, I'll just have to explain what is meant clearly enough.

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People usually just use "power" in semigroups in general. If the operation is commutative (and not written multiplicatively) "multiple" is also seen. – Henning Makholm Nov 3 '11 at 18:48
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The question is not whether the notation is appropriate, but rather if it's well defined. And (I think) the well-definition of "power" notation depends solely on whether the operation is associative (at the very least, when you restrict it to operate on $a$ multiplied by any element of the form $a\cdot a \cdot \dots \cdot a$). If that is the case, then there won't be any problem using the notation, since it unambiguously describes a specific operation.

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