R-minimal element

Let A be a set and R be a relation on A. Suppose that B is a subset of A. An element aB is said to be R-minimalPlanetmathPlanetmath in B if and only if there is no xB such that xRa. An R-minimal element in A is simply called R-minimal.

From this definition, it is evident that if A has an R-minimal element, then R is not reflexiveMathworldPlanetmathPlanetmath. However, the definition of R-minimality is sometimes adjusted slightly so as to allow reflexivity: aB is R-minimal (in B) iff the only xB such that xRa is when x=a.

Remark. Using the second definition, it is easy to see that when R is a partial orderMathworldPlanetmath, then an element a is R-minimal iff it is minimal.

Title R-minimal element
Canonical name RminimalElement
Date of creation 2013-03-22 12:42:43
Last modified on 2013-03-22 12:42:43
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 03B10
Synonym R-minimal
Synonym R-minimal
Related topic WellFoundedRelation