Differences

This shows you the differences between two versions of the page.

commutative_idempotent_integral_involutive_fl-algebras [2019/11/17 16:27] (current)
pnotthesamejipsen created
Line 1: Line 1:
 +=====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 )]
 +
 +