Table of Contents
Bounded residuated lattices
Abbreviation: RLat$_b$
Definition
A \emph{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
| Classtype | variety |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | undecidable |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence $n$-permutable | yes, $n=2$ |
| Congruence regular | yes |
| Congruence uniform | no |
| Congruence extension property | yes |
| Definable principal congruences | no |
| Equationally def. pr. cong. | no |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
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
