On semigroups of quotients and semisimplicity (Q1806089)

Semigroups with unit element are considered. Let \(A\) be a proper subset of a semigroup \(S\) which is multiplicatively closed. A right semigroup of fractions is a semigroup \(S[A^{-1}]\) together with a semigroup homomorphism \(\varphi\colon S\to S[A^{-1}]\) such that \(\varphi(a)\) is invertible for every \(a\in A\), every element in \(S[A^{-1}]\) has the form \(\varphi(s)\varphi(a)^{-1}\) with \(a\in A\) and the equality \(\varphi(s)=\varphi(t)\) implies the existence of an element \(b\in A\) such that \(sb=tb\). An element \(m\in S\) is called regular, if \(mx=my\) or \(xm=ym\) implies \(x=y\) for any elements \(x\), \(y\) of \(S\). The regular elements of \(S\) from the subsemigroup \(M\). If \(S[M^{-1}]\) exists, then it is called the right semigroup of quotients of \(S\). The existence and properties of these concepts are investigated.