FUZZY SETS AND SYSTEMS, vol.157, no.2, pp.202-204, 2006 (SCI-Expanded)
R. Belohlavek has proposed the notion of L-algebra in [R. Belohlavek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Press, New York, 2002], and a series of papers in this direction has been written by him and his colleague [R. Belohlavek, V Vychodil, Algebras with fuzzy equalities, in: Proceedings of the 10th IFSA World Congress, June 29-July 2, 2003, pp. 1-4; Algebras with fuzzy equalities, Fuzzy Sets and Systems, accepted for publication; V. Vychodil, Direct limits and reduced products of algebras with fuzzy equalities, submitted for publication]. In this short note, it will be shown that if the ordinary part of an L-algebra contains two binary operations forming the field structure, then the underlying L-equality of the L-algebra is constant (trivial). This means that L-algebras containing field structures in their ordinary parts correspond to ordinary algebras in a one-to-one manner, so all results in [R. Belohlavek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Press, New York, 2002; R. Belohlavek, V. Vychodil, Algebras with fuzzy equalities, in: Proceedings of the 10th IFSA World Congress, June 29-July 2, 2003, pp. 1-4; Algebras with fuzzy equalities, Fuzzy Sets and Systems, accepted for publication; V Vychodil, Direct limits and reduced products of algebras with fuzzy equalities, submitted for publication] in their setting for L-algebras containing field structure become trivial. In addition to this observation, other aim of this note is to draw attention to the natural question "does there exist any L-algebra with an L-equality different from trivial L-equalities in case the ordinary part of the L-algebra includes two binary operations that define group, ring, module or vector space structure?" This question is equivalent to the problem of whether the notion of L-algebra provides a meaningful generalization of ring, module or vector space structure in the context of fuzzy equality. (c) 2005 Elsevier B.V. All rights reserved.