Table of Contents
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
$x\wedge y=x\cdot y$
Examples
Example 1:
Basic results
Properties
| Equational theory | decidable |
|---|---|
| Quasiequational theory | decidable |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | yes, $n=2$ |
| Congruence e-regular | yes, $e=1$ |
| Congruence uniform | no |
| Congruence extension property | yes |
| Definable principal congruences | yes |
| Equationally def. pr. cong. | yes |
| Amalgamation property | yes |
| Strong amalgamation property | yes |
| Epimorphisms are surjective | yes |
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &2
f(5)= &3
f(6)= &5
f(7)= &8
f(8)= &15
f(9)= &26
f(10)= &47
f(11)= &82
f(12)= &151
f(13)= &269
f(14)= &494
f(15)= &891
f(16)= &1639
f(17)= &2978
f(18)= &5483
f(19)= &10006
f(20)= &18428
Values known up to size 49 [Erne, Heitzig, Reinhold (2002)]
\end{array}$
