Abbreviation: CIdInFL

A \emph{commutative idempotent involutive FL-algebra} or \emph{commutative idempotent involutive residuated lattice} is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \sim\rangle$ of type $\langle 2, 2, 2, 0, 1\rangle$ such that

$\langle A, \vee, \wedge\rangle$ is a lattice

$\langle A, \cdot, 1\rangle$ is a semilattice with top

$\sim$ is an \emph{involution}: ${\sim}{\sim}x=x$ and

$xy\le z\iff x\le {\sim}(y({\sim}z))$

A \emph{commutative involutive FL-algebra} or \emph{commutative involutive residuated lattice} is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \sim\rangle$ of type $\langle 2, 2, 2, 0, 1\rangle$ such that

$\langle A, \vee\rangle$ is a semilattice

$\langle A, \cdot\rangle$ is a semilattice and

$x\le z\iff x\cdot{\sim}y\le{\sim}1$, where $x\le y\iff x\vee y=y$.

Let $\mathbf{A}$ and $\mathbf{B}$ be involutive residuated lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \vee y)=h(x) \vee h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h({\sim}x)={\sim}h(x)$ and $h(1)=1$.

Example 1:

$\begin{array}{lr}

f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &2\\
f(5)= &2\\
f(6)= &4\\
f(7)= &4\\
f(8)= &9\\
f(9)= &10\\
f(10)= &21\\
f(11)= &22\\
f(12)= &49\\
f(13)= &52\\
f(14)= &114\\
f(15)= &121\\
f(16)= &270\\

\end{array}$

Sugihara algebras subvariety

Commutative involutive FL-algebras supervariety