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”.
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.