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.