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## FL-algebras

Abbreviation: FL

### Definition

A full Lambek algebra, or 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:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$, $h(0)=0$

Example 1:

### Properties

Classtype variety

Equational theory & decidable 1) \\\hline

Quasiequational theory undecidable undecidable no unbounded yes yes yes, n=2 no yes no no no no

### 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

### References

1) Hiroakira Ono, Yuichi Komori, Logics without the contraction rule, J. Symbolic Logic, 501985, 169–201 MRreview ZMATH implementation