## Commutative idempotent involutive FL-algebras

Abbreviation: **CIdInFL**

### Definition

A ** commutative idempotent involutive FL-algebra** or

**is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \sim\rangle$ of type $\langle 2, 2, 2, 0, 1\rangle$ such that**

*commutative idempotent involutive residuated lattice*$\langle A, \vee, \wedge\rangle$ is a lattice

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

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

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

### Definition

A ** commutative involutive FL-algebra** or

**is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \sim\rangle$ of type $\langle 2, 2, 2, 0, 1\rangle$ such that**

*commutative involutive residuated lattice*$\langle A, \vee\rangle$ is a semilattice

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

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

##### Morphisms

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 $h(1)=1$.

### Examples

Example 1:

### Basic results

### Properties

### Finite members

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

### Subclasses

Sugihara algebras subvariety

### Superclasses

Commutative involutive FL-algebras supervariety

### References

^{1)}N. Galatos and P. Jipsen,

**, Transactions of the AMS, 365 (2013), 1219-1249**

*Residuated frames with applications*