On intuitionistic fuzzy \(M\Gamma\)-subgroups of \(M\Gamma\)-groups (Q2748654)

An intuitionistic fuzzy set (IFS) of a set \(X\) is a pair \(A=(\mu_A,\gamma_A)\), where \(\mu_A\) and \(\gamma_A\colon X\to[0,1]\) are mappings such that \(\mu_A(x)+\gamma_A(x)\leq 1\) for all \(x\in X\). Let \(M\) be a \(\Gamma\)-near-ring and let \(G\) be an \(M\Gamma\)-group. An IFS \((\mu_A,\gamma_A)\) of \(G\) is called an intuitionistic fuzzy \(M\Gamma\)-subgroup of \(G\) if (IF1) \(\mu_A(x-y)\geq\min\{\mu_A(x),\mu_A(y)\}\) and \(\gamma_A(x-y)\geq\max\{\gamma_A(x),\gamma_A(y)\}\); and (IF2) \(\mu_A(a\alpha y)\geq\mu_A(y)\) and \(\gamma_A(a\alpha y)\geq\gamma_A(y)\) for all \(x,y\in G\), \(a\in M\). It is shown that intersections of intuitionistic fuzzy \(M\Gamma\)-subgroups of \(G\) are again intuitionistic fuzzy \(M\Gamma\)-subgroups, where intersections are defined in a natural way. Let \((\mu_A,\gamma_A)\) be an IFS of \(G\) and let \(a\in[0,1]\). Then the sets \(\mu^\leq_{A,a}:=\{x\in G:\mu_A(x)\geq a\}\) and \(\gamma^\geq_{A,a}:=\{x\in G:\gamma_A(x)\leq a\}\) are called a \(\mu\)-level \(a\)-cut and a \(\gamma\)-level \(a\)-cut of \(A\), respectively. It is shown that an IFS \((\mu_A,\gamma_A)\) of \(G\) is an intuitionistic fuzzy \(M\Gamma\)-subgroup of \(G\) if and only if \(\mu^\leq_{A,a}\) and \(\gamma^\geq_{A,a}\) are \(M\Gamma\)-subgroups of \(G\) for all \(a\in[0,1]\). If \(\mu\colon G\to[0,1]\) is a mapping, then \(\overline\mu\) is defined by \(\overline\mu(x)=1-\mu(x)\) for all \(x\in G\). An IFS \(A:=(\mu_A,\gamma_A)\) is an intuitionistic fuzzy \(M\Gamma\)-subgroup of \(G\) if and only if \(\square A:=(\mu_A,\overline{\mu_A})\) and \(\lozenge A:=(\overline{\gamma_A},\gamma_A)\) are intuitionistic fuzzy \(M\Gamma\)-subgroups of \(A\). Finally, some results concerning \(M\Gamma\)-homomorphisms are proved.