Bounded residuated lattices

Abbreviation: RLat$_b$

Definition

A bounded residuated lattice is a residuated lattice that is bounded:

$\bot$ is the least element: $\bot\vee x=x$

$\top$ is the greatest element: $\top\vee x=\top$

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be bounded residuated lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a residuated lattice homomorphism $h:A\rightarrow B$ that preserves the bounds: $h(\bot)=\bot$ and $h(\top)=\top$.

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