# Partial Order Relation

**DEFINITION1:** Let be a set and . If the relation is reflexive, antisymmetric and transitive, then the relation is called a "partial order relation" and denoted by in general. If "" is a partial order relation over a set , then is called "partially ordered set" or shortly "poset".

**DEFINITION2:** Let and are elements of a partially ordered set . If it holds “”, then and are called “comparable”. Otherwise they are called “incomparable”.

**DEFINITION3:** If and are comparable for all in a partially ordered set , then the relation is called a “total order” and the set is called a “totally ordered set” or “linearly ordered set”.

**DEFINITION4:** Let be a partially ordered set and . If is a totally ordered set, then is called a “chain” in .

**DEFINITION5:** Let be a partially ordered set and . If there exists an element satisfying for all , then is called the maximum of , and if there exists an element satisfying for all , then is called the minimum of . The minimum and the maximum of are denoted by and respectively.