Fuzzy maximal ideals of gamma near-rings (Q2776889)

Let \(M\) be a \(\Gamma\)-near-ring. A fuzzy set in \(M\) is called a fuzzy left (resp. right) ideal if (a) \(\mu(x-y)\geq\min\{\mu(x),\mu(y)\}\) and (b) \(\mu(u\alpha(x+v)-u\alpha v)\geq\mu(x)\) (resp. \(\mu(x\alpha u)\geq\mu(x)\)) for all \(u,v,x\in M\) and \(\alpha\in\Gamma\). If \(\mu\) is a fuzzy set, then \(M_\mu:=\{x\in M\mid\mu(x)=\mu(0)\}\). A fuzzy ideal \(\mu\) is called normal if \(\mu(0)=\mu(1)\) and complete if \(\mu(x)=0\) for some \(x\in M\). The sets of normal and complete normal fuzzy ideals of \(M\) are denoted \({\mathcal N}(M)\) and \({\mathcal C}(M)\), respectively. A fuzzy ideal \(\mu\) is called fuzzy maximal if \(\mu\) is non-constant and \(\mu^*\) is maximal in the poset \(({\mathcal N}(M),\subseteq)\) where \(\mu^*(x):=\mu(x)+1-\mu(0)\) for all \(x\in M\). It is shown that if \(\mu\) is a fuzzy maximal ideal, then (i) \(\mu\) is normal; (ii) \(\mu^*\) takes only the values 0 and 1; (iii) \(\mu_{M_\mu}=\mu\) (where \(\mu_A\) is the characteristic function of \(A\)); (iv) \(M_\mu\) is a maximal ideal of \(M\). It is then shown that every non-constant maximal element of \(({\mathcal N}(M),\subseteq)\) is also a maximal element of \(({\mathcal C}(M),\subseteq)\). Moreover, every fuzzy maximal ideal of \(M\) is complete normal.