From relation to emulation: The covering lemma for transformation semigroups (Q1916148)

A transformation semigroup here is a pair \((X,S)\) where \(X\) is a set (of states) and \(S\) is a semigroup acting on the right of \(X\). We denote by \(P^*(Z)\) the collection of nonempty subsets of a set \(Z\). Let \((X,S)\) and \((Y,T)\) be two transformation semigroups. A relational morphism \(R : (X,S) \triangleleft (Y,T)\) is a pair of functions \(\theta : X \to P^*(Y)\) and \(\varphi : S \to P^*(T)\) such that \(y \in \theta(x)\) and \(t \in \varphi(s)\) implies \(y \cdot t \in \theta(x \cdot s)\) for all \(x \in X\) and \(s \in S\). The relational morphism \(R\) is defined to be surjective if \(Y = \bigcup_{x \in X} \theta(X)\) and \(T = \bigcup_{s \in S}\varphi (s)\) and it is defined to be injective if \(\theta(x) \cap \theta(x') \neq \emptyset\) implies \(x = x'\). An injective relational morphism is referred to as an emulation or a covering. The kernel \(D_R\) of a relational morphism \(R : (X,S) \triangleleft (Y,T)\) consists of an indexed collection of sets and a collection of arrows \(\text{Arr}_{y,t}\). The indexed collection of sets is \(\{\theta^{-1}(y) : y \in \text{Im }\theta\}\) and \(\text{Arr}_{y,t} = \{[y,s,t] : y \in \text{Im }\theta\) and \(t \in \varphi(s)\}\). Two arrows \([y,s,t]\) and \([y',s',t']\) are identified if \(y = y'\), \(t = t'\) and \(x \cdot s = x \cdot s'\) for all \(x \in \theta^{-1}(y)\). The Covering Lemma is proved which involves a relational morphism and its kernel and this result turns out to have a number of applications. For example, a new, rather short, proof of the Krohn-Rhodes Theorem is produced and results about Teissier semigroups and Baer-Levi semigroups are also obtained.