Abbreviation: BLA

A basic logic algebra or 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.

A 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.

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:

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

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

MV-algebras

Heyting algebras

Generalized basic logic algebras

FLew-algebras