Brouwerian algebras

Abbreviation: BrA

Definition

A \emph{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 \emph{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

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