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Math Help - ideals of a semigroup - topology

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    ideals of a semigroup - topology

    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.
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    MHF Contributor Drexel28's Avatar
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    Re: ideals of a semigroup - topology

    Quote Originally Posted by ymar View Post
    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.
    It would seem so. If \mathcal{I} denotes the set of all ideals of the semigroup (S,*) then we have that, for any \left\{I_\alpha\right\}_{\alpha\in\mathcal{A}} \subseteq \mathcal{I}, \displaystyle S\bigcup_{ \alpha\in\mathcal{A}}I_\alpha=\bigcup_{\alpha\in \mathcal{A}}SI_\alpha=\bigcup_{\alpha\in\mathcal{A  }}I_\alpha (similarly for the right hand side) and the same idea will work for intersections and since evidently S\in\mathcal{I} if you throw in \varnothing I would agree that \mathcal{I} is a topology. A not very well behaved one though it seems.
    Last edited by Drexel28; June 14th 2011 at 01:24 PM.
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    Re: ideals of a semigroup - topology

    Hi, Drexel, thanks for your reply. I'm unable to understand your formulas though. It looks like something happened to the code and some bits disappeared.
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    Re: ideals of a semigroup - topology

    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.
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    MHF Contributor Drexel28's Avatar
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    Re: ideals of a semigroup - topology

    Quote Originally Posted by ymar View Post
    Hi, Drexel, thanks for your reply. I'm unable to understand your formulas though. It looks like something happened to the code and some bits disappeared.
    Sorry, LaTeX is acting up. I think you can figure it out from what I was able to fix.
    Quote Originally Posted by ymar View Post
    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.
    Right. I'm not really sure what it could be used for (I don't even know anyone who studies semigroups). Have you tried giving an alternate characterization of continuous maps? Are all non-constant continuous maps morphisms?
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    Re: ideals of a semigroup - topology

    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 \mathcal{J}-classes of the semigroup are exactly the classes of topological indistinguishableness.
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    MHF Contributor Drexel28's Avatar
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    Re: ideals of a semigroup - topology

    Quote Originally Posted by ymar View Post
    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.)
    Right, of course. Morphisms just is a lazy way of saying homomorphism in whatever category you are working in, so since we are dealing with semigroups it would be a semigroup homomorphism.



    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 \mathcal{J}-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?
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    Re: ideals of a semigroup - topology

    Quote Originally Posted by Drexel28 View Post
    I don't know what Green's relations are. The second part of that made sense to me though.
    There are five Green's relations and \mathcal{J} is one of them. It's the relation of generating the same principal ideal.
    Is there particular reason you are studying semigroups?
    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.
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