Brouwerian algebras

Abbreviation: BrA

Definition

A Brouwerian algebra is a structure $\mathbf{A}=\langle A, \vee, \wedge, 1, \rightarrow\rangle$ such that

$\langle A, \vee, \wedge, 1\rangle$ is a distributive lattice with top

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

Morphisms

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

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

Definition

A Brouwerian algebra is a BL-algebra $\mathbf{A}=\langle A, \vee, \wedge, 1, \cdot, \rightarrow\rangle$ such that

$x\wedge y=x\cdot y$

Example 1:

Properties

Equational theory decidable decidable undecidable no unbounded yes yes yes, $n=2$ yes, $e=1$ no yes yes yes yes yes yes

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &3\\ f(6)= &5\\ f(7)= &8\\ f(8)= &15\\ f(9)= &26\\ f(10)= &47\\ f(11)= &82\\ f(12)= &151\\ f(13)= &269\\ f(14)= &494\\ f(15)= &891\\ f(16)= &1639\\ f(17)= &2978\\ f(18)= &5483\\ f(19)= &10006\\ f(20)= &18428\\ Values known up to size 49 [Erne, Heitzig, Reinhold (2002)] \end{array}$