DOI: 10.2478/s12175-012-0047-4 Math. Slovaca 62 (2012), No. 5, 815–828 ON COMPLETE PSEUDO-BL-ALGEBRAS. A CANTOR-BERNSTEIN TYPE THEOREM Andrzej Walendziak* — Magdalena Wojciechowska-Rysiawa** (Communicated by Anatolij Dvureˇ censkij ) ABSTRACT. Pseudo-BL-algebras are a noncommutative extention of BL-al- gebras. In this paper we consider polars in pseudo-BL-algebras and the class of complete pseudo-BL-algebras. In the final section, a version of the Cantor- Bernstein theorem will be proved. c 2012 Mathematical Institute Slovak Academy of Sciences 1. Introduction BL-algebras were introduced by H´ajek [11] in 1998. Chang [2] introduced MV-algebras, which are contained in the class of BL-algebras. A noncommuta- tive extention of MV-algebras, called pseudo-MV-algebras, were introduced by Georgescu and Iorgulescu [8] and independently by Rach˚ unek [20]. A concept of pseudo-BL-algebras was firstly introduced by Georgescu and Iorgulescu in 2000 as a noncommutative generalization of BL-algebras and pseudo-MV-algebras. In [5] and [6], there were given basic properties of pseudo-BL-algebras. The pseudo-BL-algebras correspond to a pseudo-basic fuzzy logic (see [12] and [13]). Complete MV-algebras are investigated in Belluce [1] and Jakub´ ık [14]; see also [3] and [17]. In this paper we consider complete pseudo-BL-algebras and characterize some type of filters in pseudo-BL-algebras, called polars. Finally, we prove a Cantor-Bernstein type theorem. 2010 Mathematics Subject Classification: Primary 03G25, 06F05. K e y w o r d s: pseudo-BL-algebra, filter, polar, complete pseudo-BL-algebra, Cantor-Bernstein Theorem.