Proving Lattice Property in Dual Posets: A Discrete Math Question

[socratesg]

I have a question from hw, the question is stated "Show that if the poset (S,R) is a lattice then the dual poset (S,R^-1) is also a lattice"

I know by Rosen theory that the dual of a Poset is also a poset but how can i prove that it is also a lattice, what def. am i missing. Any help would be greatly apprieciated.

 

[Hurkyl]

By using the fact (S, R) is a lattice, presumably.

 

[AKG]

I'm seeing all this stuff for the first time, so don't mind the detail. (S,R) is a lattice iff for each pair of elements {a,b} in S, there is an element of S denoted sup_R{a,b} satisfying:

1) aRsup_R{a,b}, bRsup_R{a,b}
2) if s in S satisfies aRs and bRs, then sup_R{a,b}Rs

and an element of S denoted inf_R{a,b} satisfying:

1) inf_R{a,b}Ra, inf_R{a,b}Rb
2) if s in S satisfies sRa and sRb, then sRinf_R{a,b}

I assume that R^-1 is defined by aR^-1b iff bRa. If so, then we want to show that for every pair {a,b} contained in S, there is a supremum and infimum with respect to R^-1. Let's go back to the supremum in (S,R), and rewrite the conditions. So:

1) aRsup_R{a,b}, bRsup_R{a,b}
2) if s in S satisfies aRs and bRs, then sup_R{a,b}Rs

becomes

1) sup_R{a,b}R^-1a, sup_R{a,b}R^-1b
2) If s in S satisfies sR^-1a and sR^-1b, then sR^-1sup_R{a,b}

So sup_R{a,b} satisfies precisely the conditions required for it to be an infimum of {a,b} with respect to R^-1. So if (S,R) is a lattice, then for every pair {a,b}, there is a supremum s and infimum i with respect to R. (S,R^-1) is a lattice because for each {a,b} there is a supremum with respect to R^-1, that being the infimum with respect to R, namely i, and there is an infimum with respect to R^-1, that being the supremum with respect to R, namely s.

A good way to think of this is to think of S being the power set of some set X, and R being inclusion $\subseteq$. Then sup{a,b} = $a \cup b$ and inf{a,b} = $a \cap b$. R^-1 would be containment, $\supseteq$ and clearly you can see why when we switch the relation around, the intersection plays the role of the union, and the union plays the role of the intersection.