## Residuated lattices

Abbreviation: RL

### Definition

A 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:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$

Example 1:

### Properties

Classtype variety decidable 1) implementation undecidable undecidable no unbounded yes yes yes, $n=2$ no yes no no no no

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &20\\ f(5)= &149\\ f(6)= &1488\\ f(7)= &18554\\ f(8)= &295292\\ \end{array}$

### References

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1) Hiroakira Ono, Yuichi Komori, Logics without the contraction rule, J. Symbolic Logic, 50, 1985, 169–201 MRreviewZMATH