Fuzzification of \(n\)-ary groupoids (Q2729639)

By an \(n\)-ary groupoid the author means a set \(G\) with one \(n\)-ary operation \(f\colon G^n\mapsto G\), \(n\geq 2\), denoted by \(\mathcal G\). \(\mathcal G\) is called unipotent if it contains an element \(\theta\) such that \(f(x,x,\dots,x)=\theta\), \(\forall x\in G\). \(\mathcal G\) is called an \(n\)-ary semigroup if it satisfies NEWLINE\[NEWLINEf(x_1^{i-1},f(x_i^{n+i-1}),x_{n+i}^{2n-1})=f(x_1^{j-1},f(x_j^{n+j-1},x_{n+j}^{2n-1})),NEWLINE\]NEWLINE for all \(x_1,x_2,\dots,x_{2n-1}\in G\) and \(i,j\in\{1,2,\dots,n\}\). A subset \(S\) is called a \(k\)-ideal of \(\mathcal G\) if \(f(x_i^{k-1},a,x_{k+1}^n)\in S\) for all \(x_1,x_2,\dots,x_n\in G\), \(a\in S\) [\textit{F. M. Sioson}, Ann. Mat. Pura Appl., IV. Ser. 68, 161-200 (1965; Zbl 0135.03502)]. If \(S\) is a \(k\)-ideal for every \(k=1,2,\dots,n\), then it is called an ideal.NEWLINENEWLINENEWLINEThe author studies fuzzy subgroupoids, normal fuzzy subgroupoids, fuzzy ideals and Cartesian products of fuzzy subgroupoids of an \(n\)-ary groupoid \({\mathcal G}=(G,f)\). An \(n\)-ary quasigroup is defined as a groupoid \(\mathcal G\) in which for all \(x_0,x_1,\dots,x_n\in G\) and for all \(i=1,2,\dots,n\) there exists a uniquely determined \(z_i\in G\) such that \(f(x_1^{i-1},z_i,x_{i+1}^n)=x_0\). In the end, the author discusses fuzzifications of quasigroups and anti fuzzy subgroupoids.