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Partially Ordered Sets Partially ordered sets Thomas Britz and Peter Cameron November 2001 These notes have been prepared as background material for the Combinatorics Study Group talks by Professor Rafael Sorkin (Syracuse University) on the topic Discrete posets and quantum gravity, which took place in October–November 2001. 1 Binary relations We begin by taking a closer look at binary relations R X X. Figure 1 shows four of the ways in which to look at a binary relation:⊆ as a× set R, as a bipartite graph G, as a directed graph D, as an incidence matrix M, where the stars of the latter are often replaced by numbers or variables. These four ways are completely∗ equivalent but their context is quite varied. For instance, we may be interested in matchings, in which case the bipartite view would be most appropriate; or, we might be interested in paths, and the directed graph view would be more suitable; and so on. Often it is even more useful to translate one context into another. Typical instances of such a translation are when linear algebra is applied to the incidence matrix M in order to describe attributes of the three other structures. In the first two of the following examples, we assume that X contains only finitely many elements. Example 1.1 Replace each star of the matrix M with an independent variable. Then a subset A X is matched∗ in (or, a partial transversal of) the bipartite graph G if and only⊆ if the rows of M corresponding to the elements of A are linearly independent. (The set A = (ai)I X is matched in G if there is a set ⊆ B = (bi)I X such that (ai;bi) R for all i I.) ⊆ 2 2 1 Example 1.2 Replace each star of the matrix M by the integer 1. Then the (a;b)’th entry of Mk equals the number∗ of paths in D from the vertex a to the vertex b. Example 1.3 Suppose we have two relations R;S X, with corresponding inci- dence matrices M and N. Replace each star of⊆ the matrices M and N by the Boolean 1 (i.e. 1+1=1). Then M + N is the incidence∗ matrix of the relation R S. [ Set Bipartite graph (a;a); ........ ............. f a ............. a ........ ............. ............. (a;b); ............. .......... t ............. t ............. ... ........... ............. ............. b ............. b .......... ( ; ); ............. b c ......... ............. t ............. t ............. (c;c) c ............. c g t t R X X ⊆ × Incidence matrix Directed graph a b c b ................ a 0 .............. .............. ............ ............ .............. .............. .............. .............. .............. s.............. ∗ ∗ .........3..... .............. ........... t ........... ...................................... ...................................... ... ... .R.. ... b 0 0 .. ... .. ... ......................... a c ......................... ∗ I c 0 0 t t ∗ Figure 1: There are always four sides to a relation The empty set 0/ is a relation, and as sets, relations may be operated upon by complement RC, intersection , and union . Apart from these, we also introduce \ [ 2 1 the identity relation DX , the inverse relation R , and the composition operation : − ◦ DX = (x;x);x X ; 1 f 2 g R− = (y;x);(x;y) R ; R R = f(x;z);(x;y) 2 Rgand (y;z) R for some y X : ◦ 0 f 2 2 0 2 g 1 The inverse R− is easy to visualise: in terms of sets, the order of each ele- ment (x;y) in the relation R is reversed; the two parts of G are interchanged; the direction of each arc of D is reversed; and the matrix M is transposed. The com- position is not hard to visualise either. Figure 2 illustrates the composition in terms of◦ bipartite graphs. ..... ....... ..... .. ............. ..... ............. ............. ..... ............. ..... ............. ............. ..... ............. ..... ............. ............. ..... ............. ..... ............. ............. ..... ............. ..... ................... ..... ............. ..... ............. ............. ..... ............. ............. ............. ..... t ..... ............. t ............. ............. t t ..... t ..... ............. ............. ............. ..... ..... ..................... ............ ..... ... ..... ............. ..... ............. ..... ............. ..... ............. ..... ............. .................. ..... ............. ............... ..... ............. ............. ..... ..... ............. ............. ..... ..... ............. ............. ..... t..... t ............. t t ............. ..... t ..... ............. ............. ..... ..... .............. ............ ....... t t t t t R R R R 0 ◦ 0 Figure 2: Composition of binary relations on a set Example 1.4 Let R;S X be two relations on X, with corresponding incidence matrices M and N. Replace⊆ each star of the matrices M and N by the Boolean 1. If X contains only finitely many elements,∗ then M N is the incidence matrix of the relation R S. · ◦ The transitive closure R of a relation R is the relation DX R (R R) (R R R) :::: [ [ ◦ [ ◦ ◦ [ It corresponds precisely to the transitive closure of the directed graph D, that is the graph obtained by adding to D the arc (a;b) whenever D contains a path from a to b. In other words, the transitive closure R is the smallest transitive relation on S which contains R. If X contains a finite number n of elements, then by replacing each star of M by a Boolean 1, we obtain an incidence matrix of R, namely Mn. ∗ 3 2 What is a poset? The term “poset” is short for “partially ordered set”, that is, a set whose elements are ordered but not all pairs of elements are required to be comparable in the order. Just as an order in the usual sense may be strict (as <) or non-strict (as ), there are two versions of the definition of a partial order: ≤ A strict partial order is a binary relation S on a set X satisfying the conditions (R ) for no x X does (x;x) S hold; − 2 2 (A ) if (x;y) S, then (y;x) = S; − 2 2 (T) if (x;y) S and (y;z) S, then (x;z) S. 2 2 2 A non-strict partial order is a binary relation R on a set X satisfying the conditions (R+) for all x X we have (x;x) R; 2 2 (A) if (x;y) R and (y;x) R then x = y; 2 2 (T) if (x;y) R and (y;z) R then (x;z) R. 2 2 2 Condition (A ) appears stronger than (A), but in fact (R ) and (A) imply (A ). So we can− (as is usually done) replace (A ) by (A) in− the definition of a strict− partial order. Conditions (R ), (R+), (A), (T)− are called irreflexivity, reflex- ivity, antisymmetry and transitivity− respectively. We can restate these conditions 1 in terms of the identity DX , the inverse R , and the composition operation : − ◦ (R ) DX R = 0/; − \ (R+) DX R; ⊆ 1 (A) R R DX ; \ − ⊆ (T) R R R. ◦ ⊆ The two definitions of a poset are essentially the same: Proposition 2.1 Let X be a set. (a) If S is a strict partial order on X, then S DX is a non-strict partial order on X. [ (b) If R is a non-strict partial order on X, then R DX is a strict partial order on X. n (c) These two constructions are mutually inverse. 4 Exercise: Prove this proposition. Thus, a poset is a set X carrying a partial order (either strict or non-strict, since we can obtain each from the other in a canonical way). If we have to choose, we use the non-strict partial order. If R and S are corresponding non-strict and strict partial orders, we write x R y to mean (x;y) R, and x <R y to mean (x;y) S; ≤ 2 2 thus x R y holds if and only if either x <R y or x = y. (The slightly awkward ≤ notation <R means that we regard R as the name of the partial order.) If there is no ambiguity about R, we simply write x y or x < y respectively. ≤ Exercise: How many different posets are there with 3 elements? A total order is a partial order in which every pair of elements is comparable, that is, the following condition (known as trichotomy) holds: for all x;y X, exactly one of x <R y, x = y, and y <R x holds. • 2 In a poset (X;R), we define the interval [x;y]R to be the set [x;y]R = z X : x R z R y : f 2 ≤ ≤ g By transitivity, the interval [x;y]R is empty if x R y. We say that the poset is locally finite if all intervals are finite. 6≤ The set of positive integers ordered by divisibility (that is, x R y if x divides y) is a locally finite poset. ≤ 3 Preorders Sometimes we need to weaken the definition of a partial order. We say that a par- tial preorder or pseudo-order is a relation R on a set X which satisfies conditions (R) (reflexivity) and (T) (transitivity). So it is permitted that distinct elements x and y satisfy (x;y) R and (y;x) R. 2 2 Proposition 3.1 Let R be a partial preorder on X. Define a relation on X by the rule that x y if and only if (x;y);(y;x) R. Then is an equivalence∼ relation ∼ 2 ∼ on X. Moreover, if x x0 and y y0, then (x;y) R if and only if (x0;y0) R. Thus, R induces in a natural∼ way a relation∼ R on the2 set X of equivalence classes2 of X; and R is a non-strict partial order on X. The partial order obtained in this way is the canonical quotient of the partial preorder R. 5 Exercise: Prove this proposition. Exercise: How many different partial preorders are there on a set of 3 elements? Many of the definitions for posets are also valid for preorders: chains, an- tichains, upsets, downsets, minimal and maximal elements, local finiteness (see below), and so on. However, the intuition behind these definitions is sometimes different than for posets. For instance, a non-trivial finite chain does not necessar- ily have a maximal element. A less well-known characterisation of finite preorders is in terms of the inci- dence matrix M.
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