Abbreviation: GBL

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

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

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:

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

Example 1:

Generalized MV-algebras

Basic logic algebras

Distributive residuated lattices