−Table of Contents
FLe-algebras
Abbreviation: FLe
Definition
A \emph{full Lambek algebra with exchange}, or \emph{FLe-algebra}, is a FL-algebras ⟨A,∨,0,∧,T,⋅,1,∖,/⟩ such that
⋅ is commutative: x⋅y=y⋅x
Remark:
Morphisms
Let A and B be FLe-algebras. A morphism from A to B is a function h:A→B that is a homomorphism:
h(x∨y)=h(x)∨h(y), h(⊥)=⊥, h(x∧y)=h(x)∧h(y), h(⊤)=⊤, h(x⋅y)=h(x)⋅h(y), h(x∖y)=h(x)∖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
f(1)=1f(2)=1f(3)=3f(4)=16f(5)=100f(6)=794
