I need to describe a particular characteristic of the set I'm working with, but I don't know the group-theoretical terms to describe it. Can someone help?
The idea is that I have a semigroup, $G$, with certain "prime" elements, or "generating" elements: call them $\Sigma = \{s_\alpha\}_{\alpha \in S}$. I don't make any claims about the cardinality of $S$, perhaps finite, perhaps countable, perhaps uncountable. The important properties that I wish to characterize are the following:
(1) If an element of the semigroup, $w$, is $w = s_1\circ s_2\circ s_3\circ\dots\circ s_n$ (with $s_n\in\Sigma$), then $w$ cannot be expressed as the product of any other sequence of prime generating elements, and in no other order. (Obviously if $s_i = s_j$ then they may be exchanged in the product.)
(2) There are no elements of $G$, other than those which can be generated as a product of finitely many elements of $\Sigma$.
(3) Finally, I need to claim that given any $w\in G$, that I can compute the factorization. I.e., there is a decidable function which maps $w$ to its unique prime factorization.
Is what I'm describing simply the free semigroup generated by $\Sigma$, or is my requirement stricter than that?
Yes, you are describing the free semigroup over $\Sigma$.
You have to be more careful when you state your question (3). Since you are asking for decidability question, you should make precise the following points: