Table of Contents
FL-algebras
Abbreviation: FL
Definition
A \emph{full Lambek algebra}, or \emph{FL-algebra}, is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \backslash, /, 0\rangle$ of type $\langle 2,0,2,0,2,1,2,2\rangle$ such that
$\langle A, \vee, \wedge, \cdot, 1, \backslash, /\rangle$ is a residuated lattice and
$0$ is an additional constant (can denote any element).
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be FL-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$, $h(0)=0$
Examples
Example 1:
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable 1) |
| 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
$\begin{array}{lr}
f(1)= &1
f(2)= &2
f(3)= &9
f(4)= &
f(5)= &
f(6)= &
\end{array}$
Subclasses
Bounded residuated lattices subvariety
FLe-algebras subvariety
FLw-algebras subvariety
FLc-algebras subvariety
Distributive FL-algebras subvariety
Superclasses
Residuated lattices reduct
