Non-existence of non-trivial L-algebras containing field structure in the sense of Be˘lohlávek and an open question

Abstract

R. Be˘lohlávek has proposed the notion of L -algebra in [R. Be˘lohlávek, 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. Be˘lohlávek, 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. Be˘lohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Press, New York, 2002; R. Be˘lohlávek, 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.