### Table of Contents

## Generalized BL-algebras

Abbreviation: **GBL**

### Definition

A \emph{generalized BL-algebra} is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

$x\wedge y=y\cdot(y\backslash x\wedge e)$, $x\wedge y=(x/y\wedge e)\cdot y$

##### Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be generalized BL-algebras. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

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

### Examples

Example 1:

### Basic results

### Properties

### Finite members

$n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

# of algs | 1 | 1 | 2 | 5 | 10 | 23 | 49 | 111 | |||

# of si's | 1 | 1 | 2 | 4 | 9 | 19 | 42 | 97 |