Please help me to understand the topic of order relations on the set. I can't understand what a complete partially ordered set is. I want to summarize how I understand the topic of order relations on a set. But this is just a theory. How this is all in practice, I find it difficult to understand.
Let M -- is some set. On this set we can consider binary relations. An arbitrary binary relation on M is a subset of the set of ordered pairs M×M.
A binary relation R to M is called an order relation if the following conditions are met:
(i) reflexivity: ∀a∈M(a,a)∈R;
(ii) transitivity: for a,b,c∈M if (a,b)∈R and (b,c)R, then (a,c)∈R;
(iii) skew symmetry (or antisymmetry): for a,b∈M if (a,b)∈R and (b,a)∈R, then a=b.
Generally speaking, some pairs a,b∈M may not satisfy the above conditions. And in general, the order relation is called partial order. A set with a given partial order is called a partially ordered set. However, if each pair a,b∈M belong to the relation R, then it is said that a linear order is given on the set M. And such a set is called linearly ordered. Linearly ordered sets are also called monotonically ordered, or simply ordered.
In other words. Let (M,≼) be a partially ordered set M with the relation ≼. If for some a,b∈M there is a≼b or b≼a, it is said that a and b are comparable (otherwise a,b are incomparable). If all elements of the set M are comparable, then it is said that a linear order is given on M. For example, the relation "a divides b" on the set of natural numbers is a relation of partial order, but not linear, since, for example, the numbers 2,3 are incomparable. On the set of integers, the same relation will not be an order relation at all, since skew symmetry does not hold. It is clear that in any partially ordered set with the relation ≼, it is possible to distinguish (it is enough to choose comparable elements) a linearly ordered subset with the same relation. Such a subset is called a chain in a partially ordered set. For example, a chain on the set of natural numbers with a partial order relation "a divides b" is the set of all pairs of the form (1,n) -- {(1,1),(1,2),...,(1, n)}, where n∈N.
Let (M,≼) be a partially ordered set with the relation ≼, and P⊂M be its subset. An element m0∈M is called the upper bound (or upper face) of a subset P if ∀m∈P m≼m0.
A partially ordered set is called complete if each of its chains has an upper bound. For example, a partially ordered set of natural numbers with the relation "a divides b" is complete.
Is everything right?