−Table of Contents
Generalized BL-algebras
Abbreviation: GBL
Definition
A \emph{generalized BL-algebra} is a residuated lattice L=⟨L,∨,∧,⋅,e,∖,/⟩ such that
x∧y=y⋅(y∖x∧e), x∧y=(x/y∧e)⋅y
Morphisms
Let L and M be generalized BL-algebras. A morphism from L to M is a function h:L→M that is a homomorphism:
h(x∨y)=h(x)∨h(y), h(x∧y)=h(x)∧h(y), h(x⋅y)=h(x)⋅h(y), h(x∖y)=h(x)∖h(y), h(x/y)=h(x)/h(y), h(e)=e
Examples
Example 1:
Basic results
Properties
Finite members
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|
# of algs | 1 | 1 | 2 | 5 | 10 | 23 | 49 | 111 | |||
# of si's | 1 | 1 | 2 | 4 | 9 | 19 | 42 | 97 |