Find whether the following partially ordered sets are lattices, whether they have a zero and unit elements and whether they are complemented lattices:
(b) The set of all ordered pairs $(a,b)$ of real numbers, partially ordered by $\leq$, where: $(a_{1},b_{1})$$\leq$$(a_{2},b_{2}$) iff $(a_{1} < a_{2})$ or $(a_{1} = a_{2}$ & $b_{1}$$\leq$$\ b_{2}$).
I feel it is a lattice because every two elements has an infimium which is (min $(a_{1},a_{2})$, min $(b_{1},b_{2})$) and every two elements has a supremium which is (max $(a_{1},a_{2})$, max $(b_{1},b_{2}).$)
Also I feel that the zero element is (0,0) and unit element is (1,1).I do not think it is a complemented lattice but I do not know how to justify this.
Could anyone help me please?
thanks.
The order relation you describe for pairs of real numbers is usually called the lexicographic order. The resulting partially ordered set is also a lattice, but the supremum of $(1,2)$, $(2,1)$ is $(2,1)$, not $(2,2)$ and their infimum is $(1,2)$, not $(1,1)$.
In fact, this partial order is a total order: every two pairs of real numbers are comparable according to your definition of $\leq$. (Contrast this to the case of the powerset of a set, in which $\{1,2\}$ and $\{2,3\}$ are not comparable: it's neither $\{1,2\} \subseteq \{2,3\}$ nor $\{2,3\} \subseteq \{1,2\}$.)
Consequently the supremum of two pairs is the larger of the two and the infimum is the smaller of the two. Of course, if the two pairs are identical, they are both the supremum and the infimum.
It's easy to see how to extend the lexicographic ordering to tuples of real numbers.