Results 1 to 7 of 7
Like Tree2Thanks
  • 1 Post By SlipEternal
  • 1 Post By SlipEternal

Thread: poset is a lattice

  1. #1
    Newbie
    Joined
    Aug 2013
    From
    pakistan
    Posts
    22

    poset is a lattice

    Q)determine whether the poset ({1,2,4,8,16},|)
    solution :
    i made a hasse diagram of this question but i dont understand how this diagram becomes a lattice.i dont under about least upper bound and lesat lower bound ,,,can 1 is least lower bound and 16 is least upper bound
    Attached Thumbnails Attached Thumbnails poset is a lattice-hasse-diagram.bmp  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Aug 2013
    From
    pakistan
    Posts
    22

    Re: poset is a lattice

    its greates upper bound
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,776
    Thanks
    2823
    Awards
    1

    Re: poset is a lattice

    Quote Originally Posted by annie12 View Post
    Q)determine whether the poset ({1,2,4,8,16},|)
    solution :
    i made a hasse diagram of this question but i dont understand how this diagram becomes a lattice.i dont under about least upper bound and lesat lower bound ,,,can 1 is least lower bound and 16 is least upper bound
    It is the least upper bound and greatest lower bound.

    The Hasse diagram is a lattice.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Aug 2013
    From
    pakistan
    Posts
    22

    Re: poset is a lattice

    but can you explain me what is meant by least upper bound and gretest lower bound
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2010
    Posts
    3,393
    Thanks
    1351

    Re: poset is a lattice

    Let $\displaystyle A = \{1,2,4,8,16\}$. Let $\displaystyle B \subseteq A$ be a nonempty subset. The least upper bound and greatest lower bound of $\displaystyle B$ is found by a two step process. First, let $\displaystyle U = \{x \in A \mid \forall b \in B, b|x\}$ and $\displaystyle L = \{x \in A \mid \forall b \in B, x|b\}$. The set $\displaystyle U$ is the set of upper bounds of $\displaystyle B$ and the set $\displaystyle L$ is the set of lower bounds of $\displaystyle B$. To find the least upper bound and greatest lower bound, let $\displaystyle U_2 = \{x \in U \mid \forall u \in U, x|u\}$ and let $\displaystyle L_2 = \{x \in L \mid \forall l \in L, l|x\}$. If $\displaystyle U_2$ is nonempty, it must contain exactly one element. Same for $\displaystyle L_2$. If $\displaystyle U_2$ contains an element, that is the least upper bound. If $\displaystyle L_2$ contains an element, that is the greatest lower bound. In a lattice, both $\displaystyle U_2$ and $\displaystyle L_2$ will be nonempty for any choice of a nonempty subset $\displaystyle B \subseteq A$.
    Thanks from annie12
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Aug 2013
    From
    pakistan
    Posts
    22

    Re: poset is a lattice

    its difficult ,can you explain me with the lattice i have given
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Nov 2010
    Posts
    3,393
    Thanks
    1351

    Re: poset is a lattice

    Suppose $\displaystyle B = \{2\}$. Then $\displaystyle U = \{2,4,8,16\}$ since $\displaystyle 2|2, 2|4, 2|8, 2|16$ and $\displaystyle L = \{1,2\}$ since $\displaystyle 1|2, 2|2$. Then $\displaystyle U_2 = \{2\}$ (so, 2 is the least element of $\displaystyle \{2,4,8,16\}$) since $\displaystyle 2|2,2|4,2|8,2|16$, but 4 does not divide 2, 8 does not divide 2, and 16 does not divide 2. $\displaystyle L_2 = \{2\}$ (so, 2 is the greatest element of $\displaystyle \{1,2\}$) because $\displaystyle 1|2, 2|2$, but 2 does not divide 1. This means the least upper bound and greatest lower bound of 2 are both 2.

    Suppose $\displaystyle B = \{2,16\}$. Then $\displaystyle U = \{16\}$ since $\displaystyle 2|16, 16|16$, but for any other element of $\displaystyle A$, 16 would not divide it. Hence, 16 is the only upper bound of $\displaystyle \{2,16\}$. Obviously, $\displaystyle U_2 = \{16\}$, so that is the least upper bound. Then $\displaystyle L = \{1,2\}$ and $\displaystyle L_2 = \{2\}$, so 2 is the greatest lower bound.

    Suppose $\displaystyle B = \{1,4,8\}$. Then the least upper bound will be 8 and the greatest lower bound will be 1.

    Etc.
    Thanks from annie12
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. poset is a lattice
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Oct 21st 2013, 02:44 AM
  2. Poset troubles
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Sep 5th 2010, 08:56 AM
  3. Discrete maths poset
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: Aug 27th 2009, 05:13 PM
  4. give a poset that has ...
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Oct 27th 2008, 05:40 PM
  5. poset
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: Nov 9th 2006, 08:27 AM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum