Abbreviation: FL$_{ec}$

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

$\cdot$ is contractive or square-increasing: $x\le x\cdot x$

Remark:

Let $\mathbf{A}$ and $\mathbf{B}$ be FLec-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:

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

Distributive FLec-algebras

Commutative square-increasing residuated lattices

FLe-algebras