caa a22 4500 467899754 CHVBK 20180406152814.0 cr unu---uuuuu 170328e20060801xx s 000 0 eng 10.1007/s00012-006-1960-6 doi (NATIONALLICENCE)springer-10.1007/s00012-006-1960-6 On the structure of generalized BL-algebras [Elektronische Daten] [P. Jipsen, F. Montagna] Abstract.: A generalized BL - algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities $$ x\Lambda y = ((x\Lambda y)/y)y = y(y\backslash (x\Lambda y)) $$ . 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 $$ {\ell } $$ -group varieties into the lattice of varieties of integral GBLalgebras. The results of this paper also apply to pseudo-BL algebras. Birkhäuser Verlag, Basel, 2007 Generalized BL-algebras nationallicence basic logic nationallicence residuated lattices nationallicence generalized MV-algebras nationallicence Jipsen P. Department of Mathematics and Computer Science, Chapman University, 92866, Orange, CA, USA aut Montagna F. Department of Mathematics and Computer Science, University of Siena, Pian dei Mantellini, 44 53100, Siena, Italy aut algebra universalis Birkhäuser-Verlag; www.birkhauser.ch 55/2-3(2006-08-01), 227-238 0002-5240 55:2-3<227 2006 55 12 https://doi.org/10.1007/s00012-006-1960-6 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00012-006-1960-6 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Jipsen P. Department of Mathematics and Computer Science, Chapman University, 92866, Orange, CA, USA aut NATIONALLICENCE 700 1- Montagna F. Department of Mathematics and Computer Science, University of Siena, Pian dei Mantellini, 44 53100, Siena, Italy aut NATIONALLICENCE 773 0- algebra universalis Birkhäuser-Verlag; www.birkhauser.ch 55/2-3(2006-08-01), 227-238 0002-5240 55:2-3<227 2006 55 12 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer