Consider the Hasse diagram shown below on the set {a, b, c, d, e, f, g, h, i, j,

Question

Consider the Hasse diagram shown below on the set {a, b, c, d, e, f, g, h, i, j, k}. Answer the following questions (when listing elements of a solution set please list them in alphabetical order if you can do so). Please show steps involved. Thanks.

(a) What are the maximal elements.

(b) What are the minimal elements.

(c) Is there a greatest element.

(d) Is there a least element.

(e) Find all upper bounds for {d, e, k}.

(f) Find all lower bounds for {a, g, h}.

(g) Find the greatest lower bound for {a, g, h} if it exists.

(h) Use topological sort to generate a compatible total order on the set.

Explanation / Answer

(a)

maximal element =a,g,h,l

(b)

minimal element =e

(c)

as there is four maximal element, so there is no greatest element.

(d)

e is the minimal element

(e)

upper bound of d,e,k = d,k,c,f,j

(f)

lower bound of a,g,h = b,i,c,f,j,d,e,k

(g)

gratest lower bound is = e

(h)

order ={e,d,k,f,j,c,b,i,l,a,g,h}

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