On the structure of generalized BL-algebras

A generalized BL - algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities . 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 -group varieties into the lattice of varieties of integral GBLalgebras. The results of this paper also apply to pseudo-BL algebras. This paper is dedicated to Walter Taylor. Received March 7, 2005; accepted in final form July 25, 2005.