Locally closed semirings and iteration semirings. (Q1774121)

A *-semiring is an additively commutative semiring \((S,+,\cdot)\) with absorbing zero and identity 1 equipped with a star operation \(^*\colon S\to S\). If \((x+y)^*=(x^*y)^*x^*\) and \((xy)^*=1+x(yx)^*y\) for all \(x,y\in S\) then \((S,+,\cdot)\) is called a Conway semiring. An iteration semiring is a Conway semiring satisfying all so-called group-equations. Now, let \((S,+,\cdot)\) be a locally closed semiring, i.e., for every \(a\in S\) there is some integer \(k\geq 0\) such that \(1+a+\cdots+a^k=1+a+\cdots+a^{k+1}\). Then \((S,+,\cdot)\) becomes a star semiring by the star operation \(a^*=1+a+\cdots+a^k\). It is proved that every locally closed semiring is a Conway semiring. Moreover, if \((S,+,\cdot)\) is locally closed then, for any nonnegative integer \(n\), the matrix semiring \(S^{n\times n}\) is also locally closed. This implies that every locally closed semiring is an iteration semiring.