Table of Contents
Wajsberg hoops
Definition
A \emph{Wajsberg hoop} is a hoop $\mathbf{A}=\langle A, \cdot, \rightarrow, 1\rangle$ such that
$(x\rightarrow y)\rightarrow y = (y\rightarrow x)\rightarrow x$
Remark: Lattice operations are term-definable by $x\wedge y=x\cdot(x\rightarrow y)$ and $x\vee y=(x\rightarrow y)\rightarrow y$.
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be Wajsberg hoops. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y) $, $h(1)=1$
Examples
Example 1:
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | |
| First-order theory | |
| Locally finite | no |
| Residual size | |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable |
Congruence regular & yes Radim Belohlovek, \emph{On the regularity of MV-algebras and Wajsberg hoops}, Algebra Universalis, \textbf{44}, 2000, 375–377MRreview\\\hline
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &
f(4)= &
f(5)= &
f(6)= &
f(7)= &
\end{array}$
