EMBEDDING OF THE LATTICE OF IDEALS OF A RING INTO ITS LATTICE OF FUZZY IDEALS

Document Type: Research Paper

Author

Department of Mathematics, Ramjas College, University Of Delhi, Delhi, India

Abstract

We show that the lattice of all ideals of a ring $R$ can be embedded in the lattice of all its fuzzy
ideals in uncountably many ways. For this purpose, we introduce the concept of the generalized
characteristic function $\chi _{s}^{r} (A)$ of a subset $A$ of a ring $R$ for
fixed $r , s\in [0,1] $ and show that $A$ is an ideal of $R$ if, and only if, its generalized
characteristic function $\chi _{s}^{r} (A)$ is a fuzzy ideal of $R$. We also
show that the set of all generalized characteristic functions $C_{s}^{r}
(I(R))$ of the members of $I(R)$ for fixed $r , s\in [0,1] $ is a
complete sublattice of the lattice of all fuzzy ideals of $R$ and establish
that this latter lattice is generated by the union of all
its complete sublattices $C_{s}^{r} (I(R))$.

Keywords


[1] N. Ajmal and K. V. K. Thomas, The lattice of fuzzy subgroups and fuzzy normal subgroups,
Information Sci., 76 (1994), 1-11.
[2] T. Head, A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and
Systems, 73 (1995), 349-358.
[3] L. Wangjin, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (1982),
133-139.
[4] D. S. Malik and J. N. Mordeson, Radicals of fuzzy ideals, Information Sci., 65 (1992) 23,
239-252.
[5] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.
[6] A. Weinberger, Embedding lattices of fuzzy subalgebras into lattices of crisp sub-algebras,
Information Sci., 108 (1998), 51-70.