Consider the set of integrity levels L = {low, admin, kernel}, where kernel > admin > low. Furthermore consider the set of categories Cat containing Cat = {HRandAdmin(H),MarketTraders(T),IT Engineers(E)}.

Question: Name and compute the following bound in the product lattice: (admin,{E}) ∨ (kernel,{H}).

Answer: LUB, Least Upper Bound, = (kernel,{H,E})

In the above example I completely understand that kernel is the highest level and we combine the categories. However, why do I get a bad feeling that there are slightly harder examples where this simple trick does not apply?

Lastly, how would I calculate the GLB?

I am hoping to avoid maths if possible.

  • 1
    $\begingroup$ How is Cat ordered? $\endgroup$ – William Elliot Aug 17 at 23:13
  • $\begingroup$ @WilliamElliot That is the full question, no other details are included. $\endgroup$ – plagiarism Aug 17 at 23:48

Given two sequences A = (a,b,c), B = (u,v,w) you want to find the least upper bound of (b,w) and (c,u). It is (c,w).
The greatest lower bound is (b,u).
I'd explain more but as you do not want to bother with math, I won't.

Exercise. Prove, given two linear orders A,B,
that for AxB with the product order,
sup{ (a,b), (x,y) } = (sup{a,x}, sup{b,y}),
inf{ (a,b), (x,y) } = (inf{a,x}, inf{b,y}).

  • $\begingroup$ I think you made a typo as A should be equal to (a,b,c) not (a,b,f)?? I do not understand why GLB is (b,u) and not (a,u). Lastly, it helped me but I am not sure how it's related to my question. $\endgroup$ – plagiarism Aug 17 at 23:45
  • $\begingroup$ @plagiarism. Typo corrected. It is your problem with the names of the elements changed into something manageable provided Cat is linearly ordered. As Cat was not given an order, the problem becomes bogus. $\endgroup$ – William Elliot Aug 18 at 1:34
  • $\begingroup$ Why is the GLB (b,u) I thought it should be a,u??? $\endgroup$ – plagiarism Aug 18 at 11:58
  • $\begingroup$ @plagiarism Because (a,u) < (b,u ) < (b,w) and (c,u). $\endgroup$ – William Elliot Aug 18 at 22:31
  • $\begingroup$ So with GLB I do not pick the smallest (a,u) but pick one above (b,u)?? $\endgroup$ – plagiarism Aug 24 at 23:10

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