Definition:Summation

Let $$\left({S, +}\right)$$ be an algebraic structure where the operation $$+$$ is an operation derived from, or arising from, the addition operation on the natural numbers.

Let $$\left({a_1, a_2, \ldots, a_n}\right) \in S^n$$ be an ordered $n$-tuple in $$S$$.

Then the composite is called the sum of $$\left({a_1, a_2, \ldots, a_n}\right)$$, and is written:

$$\sum \limits_{j=1}^n a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$$

Alternatively:

$$\sum \limits_{1 \le j \le n} a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$$

If $$\Phi \left({j}\right)$$ is a propositional function of $$j$$, then we can write:

$$\sum \limits_{\Phi \left({j}\right)} a_j =$$ The sum of all $$a_j$$ such that $$\Phi \left({j}\right)$$ holds.