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commutative_idempotent_integral_involutive_fl-algebras [2019/11/17 16:27] (current)
pnotthesamejipsen created
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+=====Commutative idempotent involutive FL-algebras=====
+
+Abbreviation: **CIdInFL**
+
+====Definition====
+A \emph{commutative idempotent involutive FL-algebra} or \emph{commutative idempotent involutive residuated lattice} is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \sim\rangle$ of type $\langle 2, 2, 2, 0, 1\rangle$ such that
+
+$\langle A, \vee, \wedge\rangle$ is a [[lattice]]
+
+$\langle A, \cdot, 1\rangle$ is a [[semilattice]] with top
+
+$\sim$ is an \emph{involution}: ${\sim}{\sim}x=x$ and
+
+$xy\le z\iff x\le {\sim}(y({\sim}z))$
+
+====Definition====
+A \emph{commutative involutive FL-algebra} or \emph{commutative involutive residuated lattice} is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \sim\rangle$ of type $\langle 2, 2, 2, 0, 1\rangle$ such that
+
+$\langle A, \vee\rangle$ is a [[semilattice]]
+
+$\langle A, \cdot\rangle$ is a [[semilattice]] and
+
+$x\le z\iff x\cdot{\sim}y\le{\sim}1$, where $x\le y\iff x\vee y=y$.
+
+==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====
+
+$\begin{array}{lr} + f(1)= &1\\ + f(2)= &1\\ + f(3)= &1\\ + f(4)= &2\\ + f(5)= &2\\ + f(6)= &4\\ + f(7)= &4\\ + f(8)= &9\\ + f(9)= &10\\ + f(10)= &21\\ + f(11)= &22\\ + f(12)= &49\\ + f(13)= &52\\ + f(14)= &114\\ + f(15)= &121\\ + f(16)= &270\\ +\end{array}$
+
+
+====Subclasses====
+[[Sugihara algebras]] subvariety
+
+====Superclasses====
+[[Commutative involutive FL-algebras]] supervariety
+
+
+====References====
+
+[(GalatosJipsen2012> N. Galatos and P. Jipsen, \emph{Residuated frames with applications}, Transactions of the AMS, 365 (2013), 1219-1249 )]
+
+