6 Partially ordered set (mathematics)

(P,≤) is a Partially Ordered Set (poset) if binary relation ≤ on P is a partial order:

• ∀x∈P x≤x (reflexive)
• ∀x,y∈P (x≤y ∧ y≤x) → x=y (antisymmetric)
• ∀x,y,z∈P (x≤y ∧ y≤z) → x≤z (transitive)

≤ is a total order if also:

• ∀x,y∈P (x≤y ∨ y≤x)

< is a strict partial order if

• ∀x∈P ¬(x<x) (irreflexive)
• ∀x,y∈P x<y → ¬(y<x) (asymmetric)
• ∀x,y,z∈P (x<y & y<z) → x<z (transitive)