=====Lattice-ordered rings=====

Abbreviation: **LRng**
====Definition====
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: 

====Definition====
==Morphisms==
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: $f(x\vee y)=f(x)\vee f(y)$, $f(x\wedge y)=f(x)\wedge f(y)$, $f(x\cdot y)=f(x)\cdot f(y)$, $f(x+y)=f(x)+f(y)$.
====Examples====


====Basic results====
The lattice reducts of lattice-ordered rings are [[distributive lattices]].

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Congruence distributive]]  |yes, see [[lattices]] |
^[[Congruence extension property]]  | |
^[[Congruence n-permutable]]  |yes, $n=2$, see [[groups]] |
^[[Congruence regular]]  |yes, see [[groups]] |
^[[Congruence uniform]]  |yes, see [[groups]] |

^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |

====Finite members====

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

====Subclasses====
[[Commutative lattice-ordered rings]] 

====Superclasses====
[[Abelian lattice-ordered groups]] 


====References====

[(Ln19xx>
)]