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..