Generalized BL-algebras

A generalized BL-algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities x boolean AND y = ((x boolean AND y)/y)y = y(y\(x boolean AND y)). It is shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. We also observe that the idempotents in a GBL-algebra form a subalgebra of elements that commute with all other elements. Subsequently we construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. We also give equational bases for the varieties generated by such algebras. The construction provides a new way of order-embedding the lattice of f-group varieties into the lattice of varieties of integral GBL-algebras. The results of this paper also apply to pseudo-BL algebras.