Identities in Stone-Čech compactifications of semigroups (Q1806052)

Let \((S,\cdot)\) be a semigroup with discrete topology. Let \(\beta S\) denote the Stone-Čech compactification of the discrete set \(S\) then \(\beta S\) can be endowed with a semigroup multiplication \(\odot\) extending \(\cdot \). For \(x\in\beta S\), let \(\kappa (x)\) denote the least cardinal \(\alpha\) such that there exists a subset \(U\subseteq S\) of cardinality \(\alpha\) containing \(x\) in its closure. It is shown: if \(e\) is an identity of \((\beta S,\odot)\) then \(e\in S\); if \(e\in\beta S\setminus S\) is a right identity of \((\beta S,\odot)\) then there exist \(2^{2^{\kappa (e)}}\) distinct right identities of \((\beta S ,\odot)\); if there exists a right identity \(e\) of \((\beta S,\odot)\) with \(\kappa (e)=\omega_0\) then for every \(x\in\beta S\) such that the left translation of \(x\) is a continuous mapping of \(\beta S\) and \(xS\subseteq S\) we have \(x\in S\). An example is presented showing that for a Stone-Čech compactification of a non-discrete topological semigroup these statements do not hold.