weakest extension of a partial ordering

Let A be a commutative ring with partial ordering A+ and suppose that B is a ring that admits a partial ordering. If f:AB is a ring monomorphism (thus we regard B as an over-ring of A), then any partial ordering of B that can contain f(A+) will also contain the set B+B defined by


B+ is itself a partial ordering and it is called the weakest partial ordering of B that extends A+ (through f). It is called ”weakest” because this is the smallest partial ordering B+ of B that will transform f into a poring monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (i.e. a monomorphism in the categoryMathworldPlanetmath of partially ordered rings) f:(A,A+)(B,B+) (for simplicity, we abuse the symbol f here).

Title weakest extensionPlanetmathPlanetmathPlanetmath of a partial ordering
Canonical name WeakestExtensionOfAPartialOrdering
Date of creation 2013-03-22 18:51:40
Last modified on 2013-03-22 18:51:40
Owner jocaps (12118)
Last modified by jocaps (12118)
Numerical id 9
Author jocaps (12118)
Entry type Definition
Classification msc 13J30
Classification msc 13J25
Defines weakest extension of a partial ordering