Example of a semigroup satisfy some properties

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I want to find an example a finite semigroup $S$ and $K subseteq S$ satisfy the properties



  • For any $a,b in K$, we have $a,b in langle c rangle$ for some $c in S$.


  • $K$ does not hold closure property.


Thanks for any kind of help




semigroups




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    I want to find an example a finite semigroup $S$ and $K subseteq S$ satisfy the properties



    • For any $a,b in K$, we have $a,b in langle c rangle$ for some $c in S$.


    • $K$ does not hold closure property.


    Thanks for any kind of help




    semigroups




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I want to find an example a finite semigroup $S$ and $K subseteq S$ satisfy the properties



      • For any $a,b in K$, we have $a,b in langle c rangle$ for some $c in S$.


      • $K$ does not hold closure property.


      Thanks for any kind of help




      semigroups




      share|cite|improve this question











      I want to find an example a finite semigroup $S$ and $K subseteq S$ satisfy the properties



      • For any $a,b in K$, we have $a,b in langle c rangle$ for some $c in S$.


      • $K$ does not hold closure property.


      Thanks for any kind of help





      semigroups





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked 2 days ago









      user120386

      1,020717




      1,020717




















          1 Answer
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          Let $S=<mathbb Z_6,+>$ and $K=<1,5,+>$.






          share|cite|improve this answer

















          • 1




            In your example, $K$ is closed.
            – user120386
            2 days ago










          • @user120386 $1+1=2notin K$.
            – Michael Rozenberg
            2 days ago











          • You have written $K$ is generated by $1, 5$.
            – user120386
            2 days ago










          • No! $K$ is a set with operation $+$.
            – Michael Rozenberg
            2 days ago










          • Thanks, Michael for your kind help.
            – user120386
            2 days ago










          Your Answer




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          1 Answer
          1






          active

          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          0
          down vote













          Let $S=<mathbb Z_6,+>$ and $K=<1,5,+>$.






          share|cite|improve this answer

















          • 1




            In your example, $K$ is closed.
            – user120386
            2 days ago










          • @user120386 $1+1=2notin K$.
            – Michael Rozenberg
            2 days ago











          • You have written $K$ is generated by $1, 5$.
            – user120386
            2 days ago










          • No! $K$ is a set with operation $+$.
            – Michael Rozenberg
            2 days ago










          • Thanks, Michael for your kind help.
            – user120386
            2 days ago














          up vote
          0
          down vote













          Let $S=<mathbb Z_6,+>$ and $K=<1,5,+>$.






          share|cite|improve this answer

















          • 1




            In your example, $K$ is closed.
            – user120386
            2 days ago










          • @user120386 $1+1=2notin K$.
            – Michael Rozenberg
            2 days ago











          • You have written $K$ is generated by $1, 5$.
            – user120386
            2 days ago










          • No! $K$ is a set with operation $+$.
            – Michael Rozenberg
            2 days ago










          • Thanks, Michael for your kind help.
            – user120386
            2 days ago












          up vote
          0
          down vote










          up vote
          0
          down vote









          Let $S=<mathbb Z_6,+>$ and $K=<1,5,+>$.






          share|cite|improve this answer













          Let $S=<mathbb Z_6,+>$ and $K=<1,5,+>$.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered 2 days ago









          Michael Rozenberg

          86.9k1575178




          86.9k1575178







          • 1




            In your example, $K$ is closed.
            – user120386
            2 days ago










          • @user120386 $1+1=2notin K$.
            – Michael Rozenberg
            2 days ago











          • You have written $K$ is generated by $1, 5$.
            – user120386
            2 days ago










          • No! $K$ is a set with operation $+$.
            – Michael Rozenberg
            2 days ago










          • Thanks, Michael for your kind help.
            – user120386
            2 days ago












          • 1




            In your example, $K$ is closed.
            – user120386
            2 days ago










          • @user120386 $1+1=2notin K$.
            – Michael Rozenberg
            2 days ago











          • You have written $K$ is generated by $1, 5$.
            – user120386
            2 days ago










          • No! $K$ is a set with operation $+$.
            – Michael Rozenberg
            2 days ago










          • Thanks, Michael for your kind help.
            – user120386
            2 days ago







          1




          1




          In your example, $K$ is closed.
          – user120386
          2 days ago




          In your example, $K$ is closed.
          – user120386
          2 days ago












          @user120386 $1+1=2notin K$.
          – Michael Rozenberg
          2 days ago





          @user120386 $1+1=2notin K$.
          – Michael Rozenberg
          2 days ago













          You have written $K$ is generated by $1, 5$.
          – user120386
          2 days ago




          You have written $K$ is generated by $1, 5$.
          – user120386
          2 days ago












          No! $K$ is a set with operation $+$.
          – Michael Rozenberg
          2 days ago




          No! $K$ is a set with operation $+$.
          – Michael Rozenberg
          2 days ago












          Thanks, Michael for your kind help.
          – user120386
          2 days ago




          Thanks, Michael for your kind help.
          – user120386
          2 days ago












           

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