Two examples on hereditary pretorsion classes of semigroup automata (Q1345739)

A class of algebras is called a hereditary pretorsion class (HPC) if it is closed under quotients, subalgebras, coproducts and finite direct products. If \(T\) is a right linear topology on a monoid \(S\) then for a right \(S\)-automaton \(M\) define \(T(M)=\{\rho\mid\rho\) is a congruence on \(M\) and \((\rho : x) \in T\;\forall x\in M\}\) and \(t(M)=\{x \in M\mid\{(u,v)\mid xu=xv\} \in T\}\). HPCs of \(S\)-automata which are closed under direct products and injective envelopes are described. For example, it is proved that the following properties of a right linear topology \(T\) on a monoid \(S\) are equivalent: (1) For every \(S\)-automaton \(M\) and subautomaton \(N\), \(T(M) |_ N=T(N)\); (2) The set \(\{M\mid t(M)=M\}\) is closed under injective envelopes; (3) For every automaton \(M\), \(t(M)\) is essentially closed in \(M\); (4) Every injective automaton \(M\) is left split by its pretorsion subautomaton \(t(M)\).