The introduction of https://www.sciencedirect.com/science/article/pii/0001870882900482 starts with the following sentence:
The word problem for commutative semigroups is effectively decidable.
I know what a “decidable” problem or, more precisely, a “decidable” language over an alphabet means: there is an algorithm that terminates on every input and returns “yes” or “no”, depending on whether the input belongs to the language or not.
So, if the sentence were “The word problem for commutative semigroups is decidable”, I'd think this: for each (most likely, finite) alphabet $$ and each commutative semigroup $\langle G,\mathord\circ\rangle$, where $G\subseteq ^*$, there is an algorithm that, started with a string $s\in ^*$ as input, terminates and says whether $s\in G$ or not. Please correct me if I'm wrong.
Now, what the hell does effectively mean? Is there anything non-effectively decidable or ineffectively decidable? How would effectively decidable be different from ineffectively decidable or non-effectively decidable?
