Basic logic algebras

Abbreviation: BLA

Definition

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

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

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

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

prelinearity: $( x\to y) \vee ( y\to x) =1$

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

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

Definition

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

linearity: $( x\to y) \vee ( y\to x) =1$

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

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 $h:A\to B$ that is a homomorphism:

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

Example 1:

Properties

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

Finite members

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &2 f(4)= &5 f(5)= &10 f(6)= &23 f(7)= &49 f(8)= &111 \end{array}$

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