Glossary of Order Theory - R

R

• Reflecting. A function f between posets P and Q is said to reflect suprema (joins), if, for all subsets X of P for which the supremum sup{f(x): x in X} exists and is of the form f(s) for some s in P, then we find that sup X exists and that sup X = s . Analogously, one says that f reflects finite, non-empty, directed, or arbitrary joins (or meets). The converse property is called join-preserving.

• Reflexive. A binary relation R on a set X is reflexive, if x R x holds for all elements x, y in X.

• Residual. A dual map attached to a residuated mapping.

• Residuated mapping. A monotone map for which the preimage of a principal down-set is again principal. Equivalently, one component of a Galois connection.