## Brouwerian semilattices

Abbreviation: BrSlat

### Definition

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

$\langle A, \wedge, 1\rangle$ is a semilattice with identity

$\rightarrow$ 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 semilattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

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

### Definition

A \emph{Brouwerian semilattice} is a hoop $\mathbf{A}=\langle A, \cdot, 1, \rightarrow\rangle$ such that

$\cdot$ is idempotent: $x\cdot x=x$

Example 1:

### Properties

Classtype variety decidable yes unbounded yes yes yes, $n=2$ yes, $e=1$

### Finite members

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= &2 f(5)= &3 f(6)= &5 f(7)= &8 f(8)= &15 f(9)= &26 f(10)= &47 f(11)= &82 f(12)= &151 f(13)= &269 f(14)= &494 f(15)= &891 f(16)= &1639 f(17)= &2978 f(18)= &5483 f(19)= &10006 f(20)= &18428 \end{array}$

Values known up to size 49 1)

### References

1) M. Ern\'e, J. Heitzig, J. Reinhold, \emph{On the number of distributive lattices}, Electronic J. Combinatorics 9 (2002), no. 1, Research Paper 24, 23 pp.