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:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$, $h(1)=1$

Example 1:

Properties

Classtype variety decidable no yes yes

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

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= & f(4)= & f(5)= & f(6)= & f(7)= & \end{array}$