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Possible Duplicate:
Cross Product of Partial Orders

Suppose that $(L_1;≤_1)$ and $(L_2;≤_2)$ are partially ordered sets. We define a partial order $≤$ on the set $L_1 \times L_2$ in the most obvious way - we say $(a,b)≤(c,d)$ if and only if $a≤_1 c$ and $b≤_2 d$.

a) Show that if $(L_1;≤_1)$ and $(L_2;≤_2)$ are both lattices, then so is $(L_1 \times L_2;≤)$.

b) Show that if $(L_1;≤_1)$ and $(L_2;≤_2)$ are both modular lattices, then so is $(L_1 \times L_2;≤)$.

c) Show that if $(L_1;≤_1)$ and $(L_2;≤_2)$ are both distributive lattices, then so is $(L_1 \times L_2;≤)$.

d) Show that if $(L_1;≤_1)$ and $(L_2;≤_2)$ are both Boolean algebras, then so is $(L_1 \times L_2;≤)$.

I am having trouble solving this problem. Any help would be appreciated.

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    Well, show us what you did on (a) and where you got stuck. For example, what is the definition of "lattice".2012-10-11

1 Answers 1