Connection Between Triangular or Trapezoidal Fuzzy Numbers and Ideals
Paavo Kukkurainen
Starting with commutative rings R we form the ideals of R. Equipping Id(R) with ideal operations we can define semiring Sem(R) whose elements are the ideals of R. Now it is possible to define Łukasiewitz ring (Definition 2.7) which will play an essential role in this work. Then we show that quotient rings ℝ/(a) are Łukasiewitz rings where ℝ are real numbers and (a) is a principal ideal generated by a in ℝ. Using the results of L.B. Belluce and A. Di Nola concerning Łukasiewitz rings we create one to one correspondence between triangular and trapezoidal fuzzy numbers A(a1, a2, · · · , an ) and the ideals of the direct product of ℝ/(ai) (Corollary 4.3). Figures will play an important role in this case because every ℝ/(ai) has a geometrical interpretation. Let F̄ be a set of triangular and trapezoidal fuzzy numbers or fuzzy numbers which include both cases and whose membership functions intersect at least one point. Let c1, c2 · · · , ck be the common points. We denote the set of these fuzzy numbers by C. Further let Ḡ be a set of the ideals of the direct product of ℝ/(ci). We will prove that h : (F̄,∪,∩) → (Ḡ,∨,∧), h(C) = Id(ℝ/(c1) × · · · × ℝ/(ck)) is a lattice isomorphism (Proposition 4.4).
Keywords: Commutative rings, ideals, quotient rings, fields, semirings, Łukasiewitz rings, triangular and trapezoidal fuzzy numbers, direct product, lattice isomorphism