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Math Help - poset is a lattice

  1. #1
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    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
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  2. #2
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    Re: poset is a lattice

    its greates upper bound
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  3. #3
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    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.
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  4. #4
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    Re: poset is a lattice

    but can you explain me what is meant by least upper bound and gretest lower bound
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  5. #5
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    Re: poset is a lattice

    Let A = \{1,2,4,8,16\}. Let B \subseteq A be a nonempty subset. The least upper bound and greatest lower bound of B is found by a two step process. First, let U = \{x \in A \mid \forall b \in B, b|x\} and L = \{x \in A \mid \forall b \in B, x|b\}. The set U is the set of upper bounds of B and the set L is the set of lower bounds of B. To find the least upper bound and greatest lower bound, let U_2 = \{x \in U \mid \forall u \in U, x|u\} and let L_2 = \{x \in L \mid \forall l \in L, l|x\}. If U_2 is nonempty, it must contain exactly one element. Same for L_2. If U_2 contains an element, that is the least upper bound. If L_2 contains an element, that is the greatest lower bound. In a lattice, both U_2 and L_2 will be nonempty for any choice of a nonempty subset B \subseteq A.
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  6. #6
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    Re: poset is a lattice

    its difficult ,can you explain me with the lattice i have given
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  7. #7
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    Re: poset is a lattice

    Suppose B = \{2\}. Then U = \{2,4,8,16\} since 2|2, 2|4, 2|8, 2|16 and L = \{1,2\} since 1|2, 2|2. Then U_2 = \{2\} (so, 2 is the least element of \{2,4,8,16\}) since 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. L_2 = \{2\} (so, 2 is the greatest element of \{1,2\}) because 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 B = \{2,16\}. Then U = \{16\} since 2|16, 16|16, but for any other element of A, 16 would not divide it. Hence, 16 is the only upper bound of \{2,16\}. Obviously, U_2 = \{16\}, so that is the least upper bound. Then L = \{1,2\} and L_2 = \{2\}, so 2 is the greatest lower bound.

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

    Etc.
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