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wajsberg_hoops [2010/07/29 15:46] (current)
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+=====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--377[[MRreview]]\\\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====
+[[Generalized Boolean algebras]]
+
+[[MV-algebras]]
+
+====Superclasses====
+[[Hoops]]
+
+
+====References====
+
+[(Ln19xx>
+)]