An answer to Demirci's open question, a clarification of his result, and a correction of his interpretation of the result

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. Bělohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Publishers, New York, 2002; R. Bělohlávek, V. Vychodil, Algebras with fuzzy equalities, in: proceedings of the 10th IFSA world congress, June 29–July 2, 2003, pp. 1–4; R. Bělohlávek, 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 Demirci's question is positive. Second, we clarify the meaning of Demirci's 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 Demirci's interpretation of his result is mistaken and that it is not true that “all results in [R. Bělohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Publishers, New York, 2002; R. Bělohlávek, V. Vychodil, Algebras with fuzzy equalities, Fuzzy Sets and Systems, to appear; R. Bělohlávek, 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.”