Differences

—		fle-algebras [2010/07/29 15:46] (current)
Line 1:		Line 1:
		+	=====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>
		+	)]