Math 3012 Lecture 17 - Introduction to posets
Lu´ıs Pereira
Georgia Tech
October 3, 2018 Relations
Definition Given a set X, a binary relation R on the set X is a subset R of the cartesian product X × X.
Example Let X = {1, 2, 3, 4}. Below are some relations on X.
R1 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (1, 4), (4, 1), (3, 4), (4, 3)}
R2 {(1, 1), (4, 4), (1, 2), (1, 3), (2, 4), (3, 4), (1, 4)}
R3 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (3, 4), (4, 3)}
R4 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 4), (2, 4), (3, 1), (3, 4)} Reflexive relations
Definition A binary relation R ⊆ X × X is called reflexive if (x, x) ∈ R for all x ∈ X.
Example Let X = {1, 2, 3, 4}.
R1 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (1, 4), (4, 1), (3, 4), (4, 3)}
R2 {(1, 1), (4, 4), (1, 2), (1, 3), (2, 4), (3, 4), (1, 4)}
R3 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (3, 4), (4, 3)}
R4 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 4), (2, 4), (3, 1), (3, 4)}
R1, R3, R4 are reflexive, but R2 is not. Antisymmetric relations
Definition A binary relation R ⊆ X × X is called antisymmetric if (x, y) ∈ R and (y, x) ∈ R implies x = y.
Remark Equivalently, this means that when x 6= y one has that (x, y) ∈ R implies (y, x) 6∈ R .
Example Let X = {1, 2, 3, 4}.
R1 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (1, 4), (4, 1), (3, 4), (4, 3)}
R2 {(1, 1), (4, 4), (1, 2), (1, 3), (2, 4), (3, 4), (1, 4)}
R3 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (3, 4), (4, 3)}
R4 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 4), (2, 4), (3, 1), (3, 4)}
R2, R4 are antisymmetric, but R1, R3 are not. Transitive relations
Definition A binary relation R ⊆ X × X is called transitive if (x, y) ∈ R and (y, z) ∈ R implies (x, z) ∈ R.
Example Let X = {1, 2, 3, 4}.
R1 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (1, 4), (4, 1), (3, 4), (4, 3)}
R2 {(1, 1), (4, 4), (1, 2), (1, 3), (2, 4), (3, 4), (1, 4)}
R3 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (3, 4), (4, 3)}
R4 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 4), (2, 4), (3, 1), (3, 4)}
R1, R2, R4 are transitive, but R3 is not. Partial order relations
Definition A binary relation R ⊆ X × X is called a partial order relation if it is reflexive, antisymmetric and transitive.
Example Let X = {1, 2, 3, 4}.
R1 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (1, 4), (4, 1), (3, 4), (4, 3)}
R2 {(1, 1), (4, 4), (1, 2), (1, 3), (2, 4), (3, 4), (1, 4)}
R3 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1), (3, 4), (4, 3)}
R4 {(1, 1), (2, 2), (3, 3), (4, 4), (1, 4), (2, 4), (3, 1), (3, 4)}
R4 is a partial order, but R1 (not antisymmetric), R2 (not reflexive), R3 (not antisymmetric nor transitive) are not. Definition of poset
Definition A partially ordered set, abbreviated as poset, is a set P together with a relation ≤ which is a partial order on P. I.e., one has the following properties: I for any x ∈ P it is x ≤ x in P (reflexive) I if x ≤ y in P and y ≤ x in P then x = y (antisymmetric) I if x ≤ y in P and y ≤ z in P then x ≤ z (transitive)
Notation In P is a poset, we write x < y in P to mean that x ≤ y in P and x 6= y. Similarly x ≥ y in P, x > y in P mean the same as y ≤ x in P, y < x in P. Examples of posets
Example When P is a collection of sets, set x ≤ y in P to mean that x is a subset of y. In this poset
{2, 6} ≤ {2, 4, 6} {2, 4, 6} > {2, 6} {3, 7} ≥ {3, 7}
Example When P is a collection of positive integers, set x ≤ y in P to mean that x divides y evenly. In this poset
3 < 6 14 ≥ 7 17 ≥ 17
but 17 6< 1000000 Linear/total orders
Example The usual orders on the number systems N (positive integers), Z (integers), Q (rationals), R (reals) satisfy an extra condition I for all x, y, either x ≤ y in P or y ≤ x in P
Definition A partial order satisfying the extra condition above is called a linear order or a total order. Piazza poll
Question Consider the relation on {a, b, c, d} given by
{(a, a), (c, c), (d, d), (a, b), (b, c), (d, c)}
This is not a partial order. Choose the additional relations necessary to make it into a partial order.
Answers (a) just (b, b) (b)( b, b) and (a, c) (c)( b, b), (b, a), (c, b) and (c, d) (d)( b, b), (a, c), (a, d), (b, d) and (c, d) Covers in a poset Definition For x 6= y two elements of a poset P, we say that x is covered by y in P if x < y and there is no z such that x < z < y Alternatively, we also say that y covers x.
In words This means that “y is bigger than x and there is nothing in between them”.
Example For sets with inclusion, I {2, 4, 8} is covered by {2, 4, 5, 8} I {1, 7} is not covered by {1, 2, 5, 6, 7}
Example For integers with division, I 13 is covered by 91 I 16 is not covered by 160
Remark Covers don’t work well in some infinite posets like Q, R.