### Table of Contents

## FL-algebras

Abbreviation: **FL**

### Definition

A \emph{full Lambek algebra}, or \emph{FL-algebra}, is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \backslash, /, 0\rangle$ of type $\langle 2,0,2,0,2,1,2,2\rangle$ such that

$\langle A, \vee, \wedge, \cdot, 1, \backslash, /\rangle$ is a residuated lattice and

$0$ is an additional constant (can denote any element).

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be FL-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$, $h(0)=0$

### Examples

Example 1:

### Basic results

### Properties

Classtype | variety |
---|---|

Equational theory | decidable ^{1)} |

Quasiequational theory | undecidable |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

Congruence distributive | yes |

Congruence modular | yes |

Congruence n-permutable | yes, n=2 |

Congruence regular | no |

Congruence e-regular | yes |

Congruence uniform | no |

Congruence extension property | no |

Definable principal congruences | no |

Equationally def. pr. cong. | no |

Amalgamation property | |

Strong amalgamation property | |

Epimorphisms are surjective |

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &2

f(3)= &9

f(4)= &

f(5)= &

f(6)= &

\end{array}$

### Subclasses

Bounded residuated lattices subvariety

FLe-algebras subvariety

FLw-algebras subvariety

FLc-algebras subvariety

Distributive FL-algebras subvariety

### Superclasses

Residuated lattices reduct

### References

^{1)}Hiroakira Ono, Yuichi Komori, \emph{Logics without the contraction rule}, J. Symbolic Logic, \textbf{50}1985, 169–201 MRreview ZMATH implementation