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## Function rings

Abbreviation: **FRng**

### Definition

A ** function ring** (or $f$

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

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

Remark:

### Definition

##### Morphisms

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

### Examples

### Basic results

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]).

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

### Subclasses

### Superclasses

### References

Trace: » function_rings