Basic Poset Stuff

Basic Poset Definitions and Terminology

Poset is short for "finite partially ordered set." Let P be a poset. If x and y are elements of P, then we say that y covers x if y > x and there is no element u such that x < u < y. We indicate this by writing x -> y. We write w -> x,y to indicate w -> x and w -> y. We write x,y -> z to indicate x -> z and y -> z. One begins to draw a version of the order (or Hasse) diagram of P by choosing a maximal element z and depicting it with a dot. Then one represents the elements covered by z by placing dots immediately below z and connecting them to z with arcs. We say P is connected if its order diagram is connected. A filter J of P is a subset which is closed above: If x is in J, then every y in P which lies above x must also be in J. An ideal I of P is a subset which is closed below: If y is in I, then every x in P which is below y must also be in I. A convex set in P is a set which can be expressed as the intersection of a filter with an ideal. Given elements w and z in P, the interval [w,z] is the subset of P consisting of all elements u such that w <= u <= z. A sequence w, x, y, ... of elements of P forms a chain if w is covered by x, then x is covered by y, etc. Here are three special kinds of posets: A shape is an upwardly closed subset P of the strict integral fourth quadrant of the plane: Here (1,-1) is the unique maximal element, and we define (x1,-y1) > (x2,-y2) if x1 < x2 and y1 < y2. A shifted shape is an upwardly closed subset P of the integral eighth octant of the plane: Here P consists of all (x,-y) such that 1 < = -y < = x, and the order is as just defined. A tree is a connected poset P which has a unique maximal element and which is such that every other element is covered by exactly one element.

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