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 $f:A\to B$ 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

$$B^{+}:=\{\sum _{i=1}^{n}f(a_i)b_i^2:n\in \mathbb{N},a_1,\mathrm{\dots },a_n\in A^{+}\}$$

$B^{+}$ is itself a partial ordering and it is called theweakest partial ordering of $B$ that extends $A^{\mathrm{+}}$ (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) $f:(A,A^{+})\to (B,B^{+})$ (for simplicity, we abuse the symbol $f$ here).