Abbreviation: BrSlat

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 $\wedge$: $x\wedge y\leq z\Longleftrightarrow y\leq x\rightarrow z$

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:

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

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:

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

Values known up to size 49 ¹⁾

Brouwerian algebras

Semilattices with identity

Hoops