### Table of Contents

## 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 |

### Finite members

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