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Distributive residuated lattices
Abbreviation: DRL
Definition
A distributive residuated lattice is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that
$\vee, \wedge$ are distributive: $x\wedge(y\vee z) =(x\wedge y) \vee (x\wedge z)$
Remark:
Morphisms
Let $\mathbf{L}$ and $\mathbf{M}$ be distributive 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 | |
| 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
$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &20\\ f(5)= &115\\ f(6)= &7782\\ \end{array}$
Subclasses
Superclasses
References
Trace: » distributive_residuated_lattices
