Results 1 to 7 of 7

Math Help - Semigroup Theory

  1. #1
    Junior Member
    Joined
    Feb 2007
    Posts
    70

    Semigroup Theory

    I'm trying to understand a particular proof, but it has a step which isn't really given any justification. Here is the part that's giving me trouble:

    "Let I be a semiprime ideal of a semigroup S, and let d \in S-I. Letting M = \{d^n| n = 1, 2, ...\}, we obtain an m-system disjoint from I."

    For those unfamiliar with the terminology, a semiprime ideal is an ideal of a semigroup S with the property that for any a \in S, aSa \subseteq I \Longrightarrow a \in I. An m-system is a non-empty subset A of S such that a,b \in A \Longrightarrow there exists x \in S such that axb \in A.

    What I'm trying to understand is why M should be disjoint from I. I understand why it is an m-system, but I don't understand why, say, d^2 can't be in I. Any insights would be appreciated.
    Last edited by spoon737; December 28th 2009 at 12:01 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by spoon737 View Post
    I'm trying to understand a particular proof, but it has a step which isn't really given any justification. Here is the part that's giving me trouble:

    "Let I be a semiprime ideal of a semigroup S, and let d \in S-I. Letting M = \{d^n| n = 1, 2, ...\}, we obtain an m-system disjoint from I."

    For those unfamiliar with the terminology, a semiprime ideal is an ideal of a semigroup S with the property that for any a \in S, aSa \subseteq I \Longrightarrow a \in I. An m-system is a non-empty subset A of S such that a,b \in A \Longrightarrow there exists x \in S such that axb \in S.

    What I'm trying to understand is why M should be disjoint from I. I understand why it is an m-system, but I don't understand why, say, d^2 can't be in I. Any insights would be appreciated.

    What book(s) are you using? The ones I have and some web sites define a semiprime ideal I as a non-empty subset of a semigroup S s.t. s^2\in I\Longrightarrow s\in I\,,\,\,\forall x\in S, and then it is clear why M is disjoint from I ...

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2007
    Posts
    70
    I'm using Introduction to Semigroups by Mario Petrich. It' pretty old (1973), so it's not entirely unlikely that the terminology is out of date.

    The definition you gave is given in my book as the definition of a completely semiprime ideal. It makes a distinction between semiprime and completely semiprime. I agree, it's pretty trivial why M and I would be disjoint in that case, but my book only says that I is semiprime.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by tonio View Post
    What book(s) are you using? The ones I have and some web sites define a semiprime ideal I as a non-empty subset of a semigroup S s.t. s^2\in I\Longrightarrow s\in I\,,\,\,\forall x\in S, and then it is clear why M is disjoint from I ...

    Tonio
    Are S-systems defined on general semigroups, or on Monoids? The great John Howie's book only seems to define them for monoids (unless I am missing something). In monoids everything here works - the condition given by Tonio follows from spoon737's definition, and so the result holds...

    Also, can you clarify your definition of an M-system? I mean, do you mean to say axb \in S instead of, say, I?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Feb 2007
    Posts
    70
    My book's definition of an m-system is exactly as stated: given a non-empty subset A of a semigroup S, A is an m-system if for any a,b \in A there exists x \in S such that axb \in A. So yes, axb is required to be in A. I have yet to come across a definition for S-system.

    I think I've figured it out, though. After searching around a bit online, I found that my book's definition of a semiprime ideal is equivalent to the following:

    An ideal I is semiprime if for any ideal J, J^2 \subseteq I implies J \subseteq I.

    Technically, the result I found at PlanetMath was given for rings, but it was easy enough to see that this was the same for semigroups. Thinking of it this way made the result much more obvious. However, since the author makes no hint of this equivalent definition, yet he still states that M and I are disjoint in such a way that seems too trivial to require proof, I have to wonder if it really does follow immediately from his definition and I'm just missing something.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by spoon737 View Post

    I think I've figured it out, though. After searching around a bit online, I found that my book's definition of a semiprime ideal is equivalent to the following:
    i'd like to see how you "figured it out" because the claim, as you've given us, is basically false! here's a simple counter-example:
    let S = \mathbb{M}_2(\mathbb{R}), the set of 2 \times 2 matrices wih real entries. S, with matrix multiplication, is a semigroup and I=(0) is a semiprime ideal of S. let d=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}. then d \notin I but d^2 = 0 \in I.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    this is for spoon737, the OP:

    if anything in my answer is unclear for you, you should ask me in this website not asking it somewhere else! that's just not right! you (almost) copied my answer in there and, even worse,

    asked them to confirm it!! that's a big insult to me! haha ... ok, i forgive you. i just wanted to tell you that what you did was wrong.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. ideals of a semigroup - topology
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: June 14th 2011, 03:36 PM
  2. the semigroup of binary relations
    Posted in the Higher Math Forum
    Replies: 1
    Last Post: June 1st 2011, 01:07 PM
  3. inverse semigroup
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 26th 2010, 01:00 AM
  4. Semigroup with left id and inverse
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: August 26th 2008, 09:45 PM
  5. [SOLVED] Abstract algebra semigroup question
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: July 9th 2008, 09:33 PM

Search Tags


/mathhelpforum @mathhelpforum