Algebras with fuzzy equalities

In our previous study, we were interested in the equational fragment of first-order fuzzy logic [2, 3]. It turned out that the models of the equational fragment (which we call algebras with fuzzy equalities) are interesting and natural structures by themselves. The present paper provides an introductory study to the structures of equational fragment of fuzzy logic. The paper is an excerpt from our [4]. We aim at emphasizing two points: First, we want to introduce algebras with fuzzy equalities via natural examples. Then, after briefly going through the basic algebraic notions and (due to the limited scope of this paper almost only) commenting on how the results of the ordinary case generalize to fuzzy setting, we pay in more detail attention to the well known representation theorem by subdirectly irreducible algebras. We also comment on some future directions.