### Table of Contents

## 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: $x\cdot x=x$

### Examples

Example 1:

### Basic results

### Properties

### 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)}

### Subclasses

### Superclasses

### 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.