Table of Contents
Residuated lattices
Abbreviation: RL
Definition
A \emph{residuated lattice} is a structure $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ of type $\langle 2,2,2,0,2,2\rangle$ such that
$\langle L, \cdot, e\rangle$ is a monoid
$\langle L, \vee, \wedge\rangle$ is a lattice
$\backslash$ is the left residual of $\cdot$: $y\leq x\backslash z\Longleftrightarrow xy\leq z$
$/$ is the right residual of $\cdot$: $x\leq z/y\Longleftrightarrow xy\leq z$
Morphisms
Let $\mathbf{L}$ and $\mathbf{M}$ be residuated lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$
Examples
Example 1:
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable 1) implementation |
| 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 | no |
| Congruence e-regular | yes |
| Congruence uniform | no |
| Congruence extension property | no |
| Definable principal congruences | no |
| Equationally def. pr. cong. | no |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &20
f(5)= &149
f(6)= &1488
f(7)= &18554
f(8)= &295292
\end{array}$
Subclasses
Superclasses
References
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