Preorders, Partioal Orders and Equivalence Relation.
18 July 2012
Predorders are the structures which have two properties, reflexitivity and trasitivity.
So more formlly a structure S = {A,∙} (∙ is a Relation symbol) is preorder if:
1- ∙ be Reflexive, i.e ∀x ∈ A, x ∙ x
2- ∙ is Transitive, i.e ∀x,y,z ∈ A, if x ∙ y and y ∙ z then x ∙ z
if we add the antisymmetric property to the list above so we will have partial order structure.
antisymmetric means: ∀x,y ∈ A if x ∙ y and y ∙ x then x = y
if we add the symmetric property to the list above instead of antisymmetry we will get an equivalence relation.
symmetric property means: ∀x,y ∈ A if x ∙ y then y ∙ x
Above definitions summerised in the Figure bellow!
