Involutive FL-algebras

Abbreviation: InFL

Definition

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

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

$\langle A, \cdot, 1\rangle$ is a monoid

$(\sim,-)$ are an involutive pair: ${\sim}-x=x=-{\sim}x$ and

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

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 $h(1)=1$.

Definition

An involutive FL-algebra is an FL-algebra $\mathbf{A}=\langle A,\vee,\wedge,\cdot,1,\backslash,/,0\rangle$ such that

involution holds: $(0/x)\backslash 0=x=0/(x\backslash 0)$

Example 1:

Properties

Classtype Value Decidable 1) No $\infty$ Yes Yes No

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &9\\ f(5)= &21\\ \end{array}$ $\begin{array}{lr} f(6)= &101\\ f(7)= &284\\ f(8)= &1464\\ f(9)= &\\ f(10)= &\\ \end{array}$

Superclasses

FL-algebras supervariety