![]() ![]() You may find the geometric interpretation above helpful in thinking about this. A partially ordered set (A, ) is called a lattice if every pair of elements a and b in L has both a least upper bound (LUB) and a greatest lower bound. This is very straightforward to prove directly: given $p_1=\langle x_1,y_1\rangle$ and $p_2=\langle x_2,y_2\rangle$ in $\Bbb Z^+\times\Bbb Z^+$, show that if it’s not the case that $p_1\,R\,p_2$, then $p_2\,R\,p_1$. This is automatic if $\langle\Bbb Z^+\times\Bbb Z^+,R\rangle$ is a linear (or total) order, which in fact it is. In order for $\langle\Bbb Z^+\times\Bbb Z^+,R\rangle$ to be a lattice, it’s not necessary that every subset have a supremum and infimum: the requirement is that every pair of elements have a supremum and infimum. A point lattice is a regularly spaced array of points. Both restrict to the representation of finite Boolean algebras as powersets of finite sets in the finite Boolean case. ![]()
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