Abbreviation: LRng

A \emph{lattice-ordered ring} (or $\ell$\emph{-ring}) is a structure $\mathbf{L}=\langle L,\vee,\wedge,+,-,0,\cdot\rangle$ such that

$\langle L,\vee,\wedge\rangle$ is a lattice

$\langle L,+,-,0,\cdot\rangle$ is a ring

$+$ is order-preserving: $x\leq y\Longrightarrow x+z\leq y+z$

${\uparrow}0$ is closed under $\cdot$: $0\leq x,y\Longrightarrow 0\leq x\cdot y$

Remark:

Let $\mathbf{L}$ and $\mathbf{M}$ be $\ell$-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 lattice reducts of lattice-ordered rings are distributive lattices.

$\begin{array}{lr} None \end{array}$

Commutative lattice-ordered rings

Abelian lattice-ordered groups