Dunn monoid

Abbreviation: DunnMon

Definition

A 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 $h(e)=e$

Examples

Example 1:

Basic results

Properties

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

Superclasses

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.