## Dunn monoid

Abbreviation: DunnMon

### Definition

A \emph{Dunn monoid} is a commutative distributive residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \to \rangle$ such that

$\cdot$ is square-increasing: $x\le x^2$

Remark: Here $x^2=x\cdot x$. These algebras were first defined by J.M.Dunn in 1) and were named by R.K. Meyer2).

##### Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be Dunn monoids. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\to y)=h(x)\to h(y)$, and $h(e)=e$

Example 1:

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype Variety Undecidable3) Yes Yes

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

### References

1) J.M. Dunn: The Algebra of Intensional Logics, PhD thesis, University of Pittsburgh, 1966.
2) R.K. Meyer: Conservative extension in relevant implication, Studia Logica 31 (1972), 39–46.
3) A. Urquhart: The undecidability of entailment and relevant implication, J. Symbolic Logic 49 (1984), 1059–1073.