Wajsberg hoops
Definition
A 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, On the regularity of MV-algebras and Wajsberg hoops, Algebra Universalis, 44, 2000, 375–377MRreview\\\hline
Congruence uniform | |
---|---|
Congruence extension property | |
Definable principal congruences | |
Equationally def. pr. cong. | |
Amalgamation property | |
Strong amalgamation property | |
Epimorphisms are surjective |
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$
Subclasses
Superclasses
References
Trace: » wajsberg_hoops