This is an abbreviated version of the Combinatorics Study Group notes by

A binary relation R on a set X is a set of ordered pairs of elements of X, that is, a subset of X ×X. We can represent R by a matrix with rows and columns indexed by X, with (x,y) entry 1 if (x,y) ∈ R, 0 otherwise. The term “poset ” is short for “partially ordered set”, that is, a set whose ele-ments 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; (A−) if (x,y) ∈ S, then (y,x) / ∈ S; (T) if (x,y) ∈ S and (y,z) ∈ S, then (x,z) ∈ S. 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; (A) if (x,y) ∈ R and (y,x) ∈ R then x = y; (T) if (x,y) ∈ R and (y,z) ∈ R then (x,z) ∈ R. 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, reflexivity, antisymmetry and transitivity respectively. We often write x < y if (x,y) ∈ S, and x ≤ y if (x,y) ∈ R. We usually prefer the non-strict version. The two definitions are essentially the same: we get from one to the other in the obvious way, setting x ≤ y if x< y or x = y, and setting x< y if x ≤ y but x 6 = y. Thus, a poset is a set X carrying a partial order (either strict or non-strict). If there is ambiguity about R, we simply write x≤R y. A total order is a partial order in which every pair of elements is comparable, that is, the following condition (known as trichotomy) holds: The Encyclopaedia of Design Theory Posets/1 • for all x,y ∈ X, exactly one of x<R y, x = y, and y<R x holds. 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}. By transitivity, the interval [x,y]R is empty if x 6≤R y. We say that the poset is locally finite if all intervals are finite. The set of positive integers ordered by divisibility (that is, x≤R y if x divides y) is a locally finite poset. 2 Properties o