Abstract :
In his note (this issue of this journal), Demirci shows that fields with fuzzy equalities have only trivial fuzzy equalities, concludes that "Therefore, in case L-algebras contain field structure, all results in [R. Belohlavek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic\Plenum Publishers, 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; R. Belohlavek, V. Vychodil, Algebras with fuzzy equalities, Fuzzy Sets and Systems, accepted; V. Vychodil, Direct limits and reduced products of algebras with fuzzy eqalities, submitted] are evident from their classical counterparts", and asks a 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?" In our reply, we show the following. First, by presenting examples of group-based L-algebras with non-trivial L-equalities, we show that the answer to Demirciʹs question is positive. Second, we clarify the meaning of Demirciʹs result and show that it is in fact a natural generalization of the well-known classical result saying that ordinary fields do not have non-trivial congruences. Third, we argue that Demirciʹs interpretation of his result is mistaken and that it is not true that "all results in [R. B?lohl?vek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic\Plenum Publishers, New York, 2002; R. Belohlavek, V. Vychodil, Algebras with fuzzy equalities, Fuzzy Sets and Systems, to appear; R. Belohlavek, V. Vychodil, Fuzzy Equational Logic, Springer, Berlin, to appear; S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Springer, New York, 1981] are evident from their classical counterparts."
