Monounary algebras containing subalgebras with meet-irreducible congruence lattice (Q2165627)

The set of all congruence lattices \(\mathrm{Con}(A,F)\) of all algebras \((A,F)\) with fixed universe \(A\) form a lattice with respect to inclusion, denoted by \(\mathcal E_A\). Let \((A,f)\) be a monounary algebra having a connected subalgenra \(B\) containing at least three cyclic elements and such that \(\mathrm{Con}(B,f|B)\) is meet-irreducible in \(\mathcal E_B\). Several conditions guaranteeing that \(\mathrm{Con}(A,f)\) is meet-irreducible in \(\mathcal E_A\) are proved. \((A,f)\) is called connected if for every \(x,y\in A\) there exist non-negative integers \(m\) and \(n\) with \(f^m(x)=f^n(y)\). The element \(a\) of \(A\) is called cyclic if there exists some positive integer \(n\) with \(f^n(a)=a\).