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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
+)]
+
+