Abbreviation: FRng

A function ring (or $f$-ring) is a lattice-ordered ring $\mathbf{F}=\langle F,\vee,\wedge,+,-,0,\cdot\rangle$ such that

$x\wedge y=0$, $z\ge 0\ \Longrightarrow\ x\cdot z\wedge y=0$, $z\cdot x\wedge y=0$

Remark:

Let $\mathbf{L}$ and $\mathbf{M}$ be $f$-rings. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\rightarrow M$ that is a homomorphism: $f(x\vee y)=f(x)\vee f(y)$, $f(x\wedge y)=f(x)\wedge f(y)$, $f(x\cdot y)=f(x)\cdot f(y)$, $f(x+y)=f(x)+f(y)$.

The variety of $f$-rings is generated by the class of linearly ordered $\ell$-rings. This means $f$-rings are subdirect products of linearly ordered $\ell$-rings, i.e. $f$-rings are representable $\ell$-rings (see e.g. [G. Birkhoff, Lattice Theory, 1967]).

$\begin{array}{lr} Only the one-element $f$-ring. \end{array}$

Commutative function rings

Lattice-ordered rings