Abbreviation: FL$_e$
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:
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:
$h(x\vee y)=h(x)\vee h(y)$, $h(\bot )=\bot$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(\top )=\top$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$
Example 1:
| 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 |
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &16
f(5)= &100
f(6)= &794
\end{array}$