I'm really embarrassed that I'm asking this question, please be kind.
Is the set of all ideals of a semigroup a topology if we consider the empty set an ideal? It seems to be an Alexandroff topology (I know this name thanks to Deveno), but since I haven't seen this statement in any book, I'm afraid I'm making an embarrassing mistake.
Clearly, it's an ill-behaved topology, since it's usually not even T0. But at least homorphisms are continuous and surmorphisms are open. I don't know if one can use it in any way though. Nevertheless, thank you for the confirmation.
The idea occured to me today, so I haven't yet given it much thought. But no, if we take any semigroup S and a simple semigroup T with their respective ideal topologies and a function f from S into T, then f is clearly continuous. There is no reason for it to be a morphism. (I don't really know what a morphism is, because I know nothing about category theory, but I understand that not all functions are morphisms.)
One more thing is evident. Do you know what Green's relations are? Every point in a semigroup with this topology has a smallest neighbourhood, that is the principal ideal generated by this point. Therefore, the -classes of the semigroup are exactly the classes of topological indistinguishableness.
I don't know what Green's relations are. The second part of that made sense to me though. Is there particular reason you are studying semigroups?One more thing is evident. Do you know what Green's relations are? Every point in a semigroup with this topology has a smallest neighbourhood, that is the principal ideal generated by this point. Therefore, the -classes of the semigroup are exactly the classes of topological indistinguishableness.
An accident, really. I'm not a strong math student, so I was looking for an easy pro-seminar for my BA. Algebra seemed easy and it so happened that this year Jan Okniński, a major semigroup theorist, was resposible for the pro-seminar. I wrote a bachelor thesis about regular and inverse semigroups (yet to be defended) and got interested in it. I'm too inexperienced a mathematician to make any general remarks about the theory, but it's a very interesting one in my opinion. Seems to stand off a little bit, but even though the beginning of it is probably to be dated around the early 1950s, it's already quite rich in my layman's view.Is there particular reason you are studying semigroups?