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Generalized BL-algebras

Abbreviation: GBL

Definition

A generalized BL-algebra is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

$x\wedge y=y\cdot(y\backslash x\wedge e)$, $x\wedge y=(x/y\wedge e)\cdot y$

Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be generalized BL-algebras. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$

Examples

Example 1:

Basic results

Properties

Finite members

1, 1, 2, 5, 10, 23, 49, 111, …

1, 1, 2, 4, 9, 19, 42, 97, … (subdirectly irreducibles)

Subclasses

Superclasses

References