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Partially ordered set

Definition Let P be a nonempty set A binary relation R on P

that is refiexive, antisymmetńc and transitive is called a partial

order. We will denote partial order with the sign _.

Ifa set P has a partial order defined on it, P is called a partially

ordered set, and we say that _ orders P.

Ifa and b are elements ofP and a _b, then we say that a

precedes b in the sense of partial order

We can also say, that a is less than or equal to b (orb

is

greater than or equal to a) in the sense of partial order

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If a _b, but a 6= b, then we write a <b, and say that a is less

than b (orb is greater than a) in the sense of partial order s on

Examples:

1.    The "less than or eąual to” 6 relation on a set R is a partial order. So (R,6) is partially ordered set.

2.    The eąuality relation on a set A is a partial order (because

”=” is reflexive, antisymmetric and transitive). So (A,=) is

partially ordered set.