=====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====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &3\\
f(4)= &16\\
f(5)= &100\\
f(6)= &794\\
\end{array}$

====Subclasses====
[[FLew-algebras]] 

[[Distributive FLe-algebras]] 

====Superclasses====
[[Commutative residuated lattices]] 

[[FL-algebras]] 


====References====

[(Ln19xx>
)]