Glossary of Order Theory - S

S

• Saturated chain. A chain such that no element can be added between two of its elements without losing the property of being totally ordered. If the chain is finite, this means that in every pair of successive elements the larger one covers the smaller one. See also maximal chain.

• Scattered. A total order is scattered if it has no densely ordered subset.

• Scott-continuous. A monotone function f : P → Q between posets P and Q is Scott-continuous if, for every directed set D that has a supremum sup D in P, the set {fx | x in D} has the supremum f(sup D) in Q. Stated differently, a Scott-continuous function is one that preserves all directed suprema. This is in fact equivalent to being continuous with respect to the Scott topology on the respective posets.

• Scott domain. A Scott domain is a partially ordered set which is a bounded complete algebraic cpo.

• Scott open. See Scott topology.

• Scott topology. For a poset P, a subset O is Scott-open if it is an upper set and all directed sets D that have a supremum in O have non-empty intersection with O. The set of all Scott-open sets forms a topology, the Scott topology.

• Semilattice. A semilattice is a poset in which either all finite non-empty joins (suprema) or all finite non-empty meets (infima) exist. Accordingly, one speaks of a join-semilattice or meet-semilattice.

• Smallest element. See least element.

• Sperner property of a partially ordered set
• Sperner poset
• Strictly Sperner poset
• Strongly Sperner poset

• Strict order. A strict order is a binary relation that is antisymmetric, transitive, and irreflexive.

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

• Symmetric. A relation R on a set X is symmetric, if x R y implies y R x, for all elements x, y in X.