Abbreviation: DunnMon

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⁾.

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:

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.

$\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}$

commutative distributive idempotent residuated lattices subvariety

bounded Dunn monoids expansion

commutative distributive residuated lattices supervariety

square-increasing commutative residuated lattices supervariety