The Level Cardinality of Fuzzy Module Under Z-Module Homomorphism on Zm Into Zn Where gcd(m,n) is Product of Powers of Primes

Abstract

The homomorphic image of a fuzzy module over an R-module is itself a fuzzy module, yet explicit bounds on level cardinalities under specific structural constraints remain largely unexplored. In prior work, such bounds were established for Z-module homomorphisms Γ : Zm →Zn when gcd(m,n) is either a prime or the product of two distinct primes, getting maxima of 3 and 4 respectively. In this paper, we extend these results to the cases gcd(m,n) = p^s, where p is a prime and s ∈ Z+ and further to the case gcd(m,n) = p^s1₁ .p^s2₂ ... p^st_k , with distinct primes p_i and s_i ∈ Z+, i = 1,2,...t. Our results contribute to a deeper understanding of the behavior of fuzzy modules under structural mappings between cyclic modules.

Keywords: Fuzzy module homomorphism; level cardinality.

2010 Mathematics Subject Classification. 20K30; 08A72