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Some properties of idempotents of \((n,m)\)-semirings. (English) Zbl 1349.16089

Summary: In this paper, some properties of the additive and multiplicative idempotents of an \((n,m)\)-semiring are investigate. Unlike the usual semiring for \(n,m\geq 3\), we show that there is additively cancellative \((n,m)\)-semiring which has more additive idempotents. They are neutral elements of the \(n\)-ary operation, too. It also shows that there are multiplicatively cancellative \((n,m)\)-semiring which has at least two multiplicative neutral elements. In addition, we give some properties of subtractive ideal of \((n,m)\)-semiring, notion introduced by Y. Zhu [Algebra 2013, Article ID 272104 (2013; Zbl 1331.16042)], as an extension of the similar definition of usual semiring.

MSC:

16Y99 Generalizations
20N15 \(n\)-ary systems \((n\ge 3)\)
17A42 Other \(n\)-ary compositions \((n \ge 3)\)
16Y60 Semirings

Citations:

Zbl 1331.16042
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