# Differences

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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>
+)]

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