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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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