Preorderings on semigroups and semirings of right quotients (Q1976429)

Let \((S,\cdot)\) be a semigroup and \(\Pi_n(S)\) the set of all finite products \(z\) in \(S\) such that every factor \(s\in S\) of \(z\) has a multiplicity which is divisible by \(n\). A subsemigroup \(T\) of \(S\) is called a preordering (of exponent \(n\)) if \(\Pi_n(S)\subseteq T\) for some integer \(n\). Now, let \(Q=Q_r(S,\Sigma)\) be the semigroup of right quotients of \(S\) with respect to a subsemigroup \(\Sigma\) of \(S\). A preordering \(T\) of \(S\) is called \(\Sigma\)-closed if \(\xi a\eta\in T\) implies \(a\in T\) for every \(a\in S\) and \(\xi,\eta\in\Sigma\cap T\). It is shown that every preordering \(T\) of \(S\) that has an extension to a preordering \(T'\) of \(Q\) is necessarily \(\Sigma\)-closed. Moreover, if such a preordering \(T'\) exists then it equals \(T(T\cap\Sigma)^{-1}\). A sufficient condition for \(T'\) to be a preordering of \(Q\) is (1) \(a\xi c\in T\) implies \(ac\in T\) for every \(a,c\in S\) and \(\xi\in T\cap\Sigma\). The necessary and the sufficient condition coincide if \(T\) has the permutation property, i.e., \(s_1\cdots s_k\in T\) iff \(s_{\pi(1)}\cdots s_{\pi(k)}\in T\) for every permutation \(\pi\). Corresponding results are proved for semirings \((S,+,\cdot)\) and semirings \(Q_r(S,\Sigma)\) of right quotients.