### Table of Contents

## FLe-algebras

Abbreviation: **FL$_e$**

### Definition

A \emph{full Lambek algebra with exchange}, or \emph{FLe-algebra}, is a FL-algebras $\langle A, \vee, 0, \wedge, T, \cdot, 1, \backslash, /\rangle$ such that

$\cdot$ is commutative: $x\cdot y=y\cdot x$

Remark:

##### Morphisms

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

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

### Examples

Example 1:

### Basic results

### Properties

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

Equational theory | decidable |

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)= &1

f(3)= &3

f(4)= &16

f(5)= &100

f(6)= &794

\end{array}$