=====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 [(Du1966)] and were named by R.K. Meyer[(Me1972)].

==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.

^[[Classtype]]                        | Variety |
^[[Equational theory]]                | Undecidable[(Ur1984)] |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | Yes |
^[[Congruence modular]]               | Yes |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====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====

[(Du1966>
J.M. Dunn: The Algebra of Intensional Logics, PhD thesis, University of Pittsburgh, 1966.
)]
[(Me1972>
R.K. Meyer: Conservative extension in relevant implication, Studia Logica 31 (1972), 39–46.
)]
[(Ur1984>
A. Urquhart: The undecidability of entailment and relevant implication, J. Symbolic Logic 49 (1984), 1059–1073.
)]