### Table of Contents

## 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. Meyer^{2)}.

##### 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 $h(e)=e$

### Examples

Example 1:

### Basic results

### 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.

### 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

commutative distributive idempotent residuated lattices subvariety

bounded Dunn monoids expansion

### Superclasses

commutative distributive residuated lattices supervariety

square-increasing commutative residuated lattices supervariety

### 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.