| Summary: | The concept of <i>j</i>-hyperideals, for all positive integers <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, in <i>n</i>-ary semihypergroups, is a generalization of the concept of left, lateral and right hyperideals in ternary semihypergroups. In this paper, we first introduce the concept of <i>j</i>-(0-)simple <i>n</i>-ary semihypergroups and discuss their related properties through terms of <i>j</i>-hyperideals. Furthermore, we characterize the minimality and maximality of <i>j</i>-hyperideals in <i>n</i>-ary semihypergroups and establish the relationships between the (0-)minimal, maximal <i>j</i>-hyperideals and the <i>j</i>-(0-)simple <i>n</i>-ary semihypergroups. Our main results are to extend and generalize the results on semihypergroups and ternary semihypergroups. Moreover, a related question raised by Petchkaew and Chinram is solved.
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