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## Basic logic algebras

Abbreviation: BLA

### Definition

A basic logic algebra or BL-algebra is a structure $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle$ such that

$\left\langle A,\vee ,0,\wedge ,1\right\rangle$ is a bounded lattice

$\left\langle A,\cdot ,1\right\rangle$ is a commutative monoid

$\rightarrow$ gives the residual of $\cdot$: $x\cdot y\leq z\Longleftrightarrow y\leq x\rightarrow z$

linearity: $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$

BL: $x\cdot(x\rightarrow y)=x\wedge y$

Remark: The BL identity implies that the lattice is distributive.

### Definition

A basic logic algebra is a FLe-algebra $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle$ such that

linearity: $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$

BL: $x\cdot (x\rightarrow y)=x\wedge y$

Remark: The BL identity implies that the identity element $1$ is the top of the lattice.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be basic logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

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

Example 1:

### Properties

Classtype variety decidable no unbounded yes yes yes, $n=2$ yes, $e=1$ no yes no

### Finite members

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

The number of subdirectly irreducible BL-algebras of size $n$ is $2^{n-2}$.

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