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lukasiewicz_algebras_of_order_n [2010/07/29 15:46] (current)
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+=====Lukasiewicz algebras of order n=====
+Abbreviation: **LA$_n$**
+====Definition====
+A \emph{Lukasiewicz algebra of order $n$} is a structure $\mathbf{A}=\langle A,\vee +,0,\wedge ,1,\neg,\sigma_0,\ldots,\sigma_{n-1}\rangle$ such that
+
+
+$\langle A,\vee ,0,\wedge ,1, \neg\rangle$ is a [[De Morgan algebras]]
+
+
+1.
+$\sigma_i$ is a lattice homomorphism:  $\sigma_i(x\vee y)=\sigma_i(x)\vee\sigma_i(y) +\mbox{and} \sigma_i(x\wedge y)=\sigma_i(x)\wedge\sigma_i(y)$
+
+2.
+$\sigma_i(x) \vee \neg(\sigma_i(x)) = 1$, $\sigma_i(x) \wedge \neg(\sigma_i(x)) = 0$
+
+3.
+$\sigma_i(\sigma_j(x)) = \sigma_j(x)$ for $1 \le j \le n-1$
+
+4.
+$\sigma_i(\neg x) = \neg(\sigma_{n-i}(x))$
+
+5.
+$\sigma_i(x) \wedge \sigma_j(x) = \sigma_i(x)$ for $i \le j \le n - 1$
+
+6.
+$x \vee \sigma_{n-1}(x) = \sigma_{n-1}(x)$, $x \wedge \sigma_1(x) = \sigma_1(x)$
+
+7.
+$y \wedge (x \vee \neg(\sigma_i(x)) \vee \sigma_{i+1}(y)) = y$ for $i \ne n - 1$
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be Lukasiewicz algebras of order $n$. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
+homomorphism:
+
+$h(x\vee y)=h(x)\vee h(y)$, $h(\neg x)=\neg h(x)$, $h(\sigma_i(x))=\sigma_i(h(x))$ for $i=0,\ldots,n-1$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+====Properties====
+^[[Classtype]]  |Variety |
+^[[Equational theory]]  |decidable |
+^[[Quasiequational theory]]  | |
+^[[First-order theory]]  | |
+^[[Congruence distributive]]  |Yes |
+^[[Congruence modular]]  |Yes |
+^[[Congruence n-permutable]]  | |
+^[[Congruence regular]]  | |
+^[[Congruence uniform]]  | |
+^[[Congruence extension property]]  | |
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]  | |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+^[[Locally finite]]  |yes |
+^[[Residual size]]  |$n$ |
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &\\ +f(3)= &\\ +f(4)= &\\ +f(5)= &\\ +f(6)= &\\ +f(7)= &\\ +f(8)= &\\ +f(9)= &\\ +f(10)= &\\ +\end{array}$
+
+====Subclasses====
+[[Boolean algebras]]
+
+====Superclasses====
+[[De Morgan algebras]]
+
+
+====References====
+
+[(Ln19xx>
+)]