Glossary of Order Theory - I

I

• An ideal is a subset X of a poset P that is a directed lower set. The dual notion is called filter.

• The incidence algebra of a poset is the associative algebra of all scalar-valued functions on intervals, with addition and scalar multiplication defined pointwise, and multiplication defined as a certain convolution; see incidence algebra for the details.

• Infimum. For a poset P and a subset X of P, the greatest element in the set of lower bounds of X (if it exists, which it may not) is called the infimum, meet, or greatest lower bound of X. It is denoted by inf X or X. The infimum of two elements may be written as inf{x,y} or x ∧ y. If the set X is finite, one speaks of a finite infimum. The dual notion is called supremum.

• Interval. For two elements a, b of a partially ordered set P, the interval is the subset {x in P | a ≤ x ≤ b} of P. If a ≤ b does not hold the interval will be empty.

• Interval finite poset. A partially ordered set P is interval finite if every interval of the form {x in P | x ≤ a} is a finite set.

• Inverse. See converse.

• Irreflexive. A relation R on a set X is irreflexive, if there is no element x in X such that x R x.

• Isotone. See monotone.