Would it be appropriate to use "power" to describe an $n$-fold fold of a number with some associative binary operation? Is there a better expression?

Question

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.

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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.

Answer 1

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.