lattices and their properties

Question

If $X$ and $Y$ are complemented lattices then how can I show that their cartesian product is complemented lattice too. I know that as every element of $X$ and $Y$ has complement then how to show that their cartesian product also has complements and it is bounded?

Answer 1

If $x^c$ serves as complement of $x$ and $y^c$ serves as complement of $y$ then: $$\langle x,y\rangle\wedge\langle x^c,y^c\rangle=\langle x\wedge x^c,y\wedge y^c\rangle=\langle0,0\rangle$$

and: $$\langle x,y\rangle\vee\langle x^c,y^c\rangle=\langle x\vee x^c,y\vee y^c\rangle=\langle1,1\rangle$$

Furter $\langle0,0\rangle$ serves as initial element of $X\times Y$ and $\langle1,1\rangle$ serves as teminal element of $X\times Y$.