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integral_involutive_fl-algebras [2016/11/18 16:01] (current)
jipsen created
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 +=====Integral involutive FL-algebras=====
 +
 +Abbreviation: **IInFL**
 +
 +====Definition====
 +An \emph{integral involutive FL-algebra} or \emph{integral involutive residuated lattice} is an [[involutive residuated lattice]]  that is
 +
 +integral: $x\vee 1 = 1$
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be involutive residuated lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
 +$h(x \vee y)=h(x) \vee h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h({\sim}x)={\sim}h(x)$ and $h(1)=1$.
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +
 +^[[Classtype]]                        |Value  |
 +^[[Equational theory]]                |Decidable [(GalatosJipsen2012)] |
 +^[[Quasiequational theory]]           | |
 +^[[First-order theory]]               | |
 +^[[Locally finite]]                   |No |
 +^[[Residual size]]                    |$\infty$ |
 +^[[Congruence distributive]]          |Yes |
 +^[[Congruence modular]]               |Yes |
 +^[[Congruence $n$-permutable]]        | |
 +^[[Congruence regular]]               | |
 +^[[Congruence uniform]]               | |
 +^[[Congruence extension property]]    | |
 +^[[Definable principal congruences]]  | |
 +^[[Equationally def. pr. cong.]]      |No |
 +^[[Amalgamation property]]            | |
 +^[[Strong amalgamation property]]     | |
 +^[[Epimorphisms are surjective]]      | |
 +
 +====Finite members====
 +
 +$\begin{array}{lr}
 +  f(1)= &1\\
 +  f(2)= &1\\
 +  f(3)= &1\\
 +  f(4)= &3\\
 +  f(5)= &3\\
 +\end{array}$    
 +$\begin{array}{lr}
 +  f(6)= &12\\
 +  f(7)= &17\\
 +  f(8)= &78\\
 +  f(9)= &\\
 +  f(10)= &\\
 +\end{array}$
 +
 +
 +====Subclasses====
 +[[Cyclic integral involutive FL-algebras]] subvariety
 +
 +====Superclasses====
 +[[Involutive FL-algebras]] supervariety
 +
 +
 +====References====
 +
 +[(GalatosJipsen>
 +N. Galatos and P. Jipsen, \emph{Residuated frames with applications}, Transactions of the AMS, to appear
 +)]
 +
 +