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Involutive residuated lattices

Abbreviation: InRL

Definition

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

$\langle A, \vee, \wedge, \neg\rangle$ is an involutive lattice

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

$xy\le z\iff x\le \neg(y(\neg z))\iff y\le \neg((\neg z)x)$

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: $h(x \vee y)=h(x) \vee h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h({\tilde}x)={\tilde}h(x)$ and $h(1)=1$.

Definition

An is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

Finite members

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

Subclasses

... subvariety

... expansion

Superclasses

... supervariety

... subreduct

References


1) F. Lastname, Title, Journal, 1, 23–45 MRreview