The structure of ordered sets

Let M be a set ordered by some relation of order A , and Q its some subset Q Ì M. Then the Majo Ranta (upper bound) of the Q subset Ì M is called such an element M Î M that for all Q Î Q , the ratio Q A M is valid.

For example: Let M be the set of natural numbers from 1 to 10, i.e. M = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and Q is a subset of Q = {5, 6, 7}. If A is a relation of order £, then the majorant of the subset Q Ì M is an element M Î M for which the condition 5 £ M ; £ 6 M ; £ 7 M. Such an element is, for example, M = 8 (as well as M = 9 and M = 10).

Minorant (lower bound) Subsets Q Ì M is called such an element M Î M , when for all Q Î Q is the ratio M A Q.

For example: For the conditions of the previous example, the minorant is such an element M Î M for which the relations M are satisfied £ 5; M £ 6; M £ 7. Such an element is, for example, M = 4 (and also M = 3; M = 2; M = 1).

The set Q Ì M can have many majorants and minorants.

If the set of majorants has a minimum, then this element is unique. It is called the Upper bound or the supremum of the Set Q and is denoted by Sup Q.

If the set of minorants has a maximum, then this element is unique. It is called the lower bound or the infinite of the set Q and is denoted by Inf Q.

If the majorant (or minorant) belongs to the set Q , then it is called the maximum (or minimum) of the set Q - max Q (min Q ). The maximum, like the minimum, is the only one (if it exists). Therefore, when they talk about the maximum or minimum, this element is the only one and quite specific.

For our example, Q = 7 is max Q , and Q = 5 is min Q.