A pair \((X,\mathcal F)\) where \(X\) is a set and \(\mathcal F\) is a set of mappings from \(X\) into itself closed under composition and the identity mapping is called an \(M\)-set. An equivalence \(\rho \) on \(X\) is called a congruence of \((X,\mathcal F)\) if for every \(x,y\in X\) with \(x\rho y\) and every \(f\in\mathcal F\) we have \(f(x)\rho f(y)\). The set of all congruences of an \(M\)-set ordered by inclusion forms an algebraic lattice.
J. Tůma [J. Algebra 125, No. 2, 367-399 (1989; Zbl 0679.20024)] proved that for every algebraic lattice \(L\) there exists an \(M\)-set \((X,\mathcal F)\) such that every \(f\in\mathcal F\) is a bijection, \(f^{-1}\in\mathcal F\) for every \(f\in\mathcal F\), for every \(x,y\in X\) there exists \(f\in\mathcal F\) with \(f(x)=y\), and the lattice of all congruences of \((X,\mathcal F)\) is isomorphic to \(L\). An \(M\)-set \((X,\mathcal F)\) is flat if there exists a decomposition \(\{X_i\mid i\in I\}\) of \(X\) such that \(f(X_i)\subseteq X_i\) for every \(i\in I\) and \(f\in\mathcal F\), and for every \(i\in I\) and \(x,y\in X_ i\) there exist \(f,g\in\mathcal F\) with \(f(x)=y\) and \(g(y)=x\). The classes \(X_i\), \(i\in I\), are called orbits of \((X,\mathcal F)\). If an \(M\)-set has at least two orbits then it is intransitive.
The aim of the paper is to study algebraic lattices that are isomorphic to the lattice of all congruences of an intransitive flat \(M\)-set. For an algebraic lattice \(L\) a special sublattice \(\Pi(L)\) is defined and a special \(\Pi\)-product of lattices is defined. It is proved that an algebraic lattice \(L\) is isomorphic to the lattice of all congruences of an intransitive flat \(M\)-set \((X,\mathcal F)\) if and only if \(\Pi(L)\) is a \(\Pi\)-product of \(\Pi(L_i)\) where \(i\) is taken over all orbits of \((X,\mathcal F)\) and \(L_i\) is isomorphic to the lattice of all congruences of the orbit \(i\). If \((X,\mathcal F)\) has exactly two orbits then \(L\) is isomorphic to \(L_1\times L_2\) where \(L_i\) is isomorphic to the lattice of all congruences of the orbit \(i\), \(i=1,2\). Also some forbidden configurations in \(L\) are found. Finally, several interesting open problems are given.