Cyclic semirings with idempotent noncommutative addition. (Q1930227)

Let \((S,+,\cdot)\) be a semiring with identity 1 and non-commutative addition. Define \(S^*=S\setminus\{0\}\) if \((S,+,\cdot)\) has an absorbing zero 0, and \(S^*=S\) otherwise. Then \((S,+,\cdot)\) is called cyclic if \(S^*\subseteq (a)=\{a^n\mid n\in\mathbb N_0\}\) for some \(a\neq 1\) in \(S\). It is proved that for an infinite cyclic semiring with non-commutative addition, \((S,+)\) is either a left zero semigroup or a right zero semigroup. A finite cyclic semiring \((S,+,\cdot)\) is called of type \((n,k)\) for some \(k\in\mathbb N_0\) and \(n\in\mathbb N\) if \(S=\{1,a,\dots,a^k,\dots,a^{k+n-1}\}\), \(|S|=n+k\geq 2\) and \(a^{n+k}=a^k\). Now, let \(S=(a)\) be a finite cyclic semiring of type \((n,k)\) with idempotent and non-commutative addition. If \(a^k\) is an absorbing element then \((S,+)\) is either a left zero semigroup or a right zero semigroup. If \(k\geq 1\), \(n\geq 2\) and \((S,+)\) is neither left zero nor right zero then \(T=\{1,a^n,\dots,a^{nl}\}\) is a cyclic subsemiring of \(S\) with commutative addition and absorbing element \(a^{nl}\).