# 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 $f:A\rightarrow B$ 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 $B^{+}\subset B$ defined by

 $B^{+}:=\{\sum_{i=1}^{n}f(a_{i})b_{i}^{2}:n\in\mathbb{N},a_{1},\dots,a_{n}\in A% ^{+}\}$

$B^{+}$ 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 monomorphism       (i.e. a monomorphism in the category  of partially ordered rings) $f:(A,A^{+})\rightarrow(B,B^{+})$ (for simplicity, we abuse the symbol $f$ here).

Title weakest extension   of a partial ordering WeakestExtensionOfAPartialOrdering 2013-03-22 18:51:40 2013-03-22 18:51:40 jocaps (12118) jocaps (12118) 9 jocaps (12118) Definition msc 13J30 msc 13J25 weakest extension of a partial ordering