FLe-algebras

Abbreviation: FL$_{ec}$

Definition

A full Lambek algebra with exchange and contraction, or 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:

Morphisms

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:

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

Examples

Example 1:

Basic results

Properties

Finite members

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

Subclasses

Superclasses

References