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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>
 +)]