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fle-algebras [2010/07/29 15:46] (current)
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 +=====FLe-algebras=====
 +Abbreviation: **FL$_e$**
 +====Definition====
 +A \emph{full Lambek algebra with exchange}, or \emph{FLe-algebra}, is a [[FL-algebras]]
 +$\langle A, \vee, 0, \wedge, T, \cdot, 1, \backslash, /\rangle$ such that
 +
 +
 +$\cdot$ is commutative:  $x\cdot y=y\cdot x$
 +
 +
 +Remark:
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be FLe-algebras. A morphism from $\mathbf{A}
 +$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
 +
 +$h(x\vee y)=h(x)\vee h(y)$, $h(\bot )=\bot$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(\top )=\top$,
 +$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  |decidable |
 +^[[Quasiequational theory]]  |undecidable |
 +^[[First-order theory]]  |undecidable |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +^[[Congruence distributive]]  |yes |
 +^[[Congruence modular]]  |yes |
 +^[[Congruence n-permutable]]  |yes, $n=2$ |
 +^[[Congruence regular]]  |no |
 +^[[Congruence e-regular]]  |yes |
 +^[[Congruence uniform]]  |no |
 +^[[Congruence extension property]]  |no |
 +^[[Definable principal congruences]]  |no |
 +^[[Equationally def. pr. cong.]]  |no |
 +^[[Amalgamation property]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &1\\
 +f(3)= &3\\
 +f(4)= &16\\
 +f(5)= &100\\
 +f(6)= &794\\
 +\end{array}$
 +
 +====Subclasses====
 +[[FLew-algebras]]
 +
 +[[Distributive FLe-algebras]]
 +
 +====Superclasses====
 +[[Commutative residuated lattices]]
 +
 +[[FL-algebras]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]