Investigation of subalgebra lattices by means of Hasse constants. (Q2496154)

Let \(A\), \(B\), \(C\) be finite algebras, \(A\) a subalgebra of \(C\). The following three numbers are called the Hasse constants: (1) the number of copies of \(B\) in \(C\), (2) the number of copies of \(B\) in \(C\) containing \(A\), (3) the number of copies of \(B\) in \(C\) containing \(A\) and being in the same automorphism orbit as \(B\). The basic properties of Hasse constants are described to facilitate the connection between the subalgebra lattice \(\text{Sub}(C)\) and the natural action of the automorphism group \(\Aut(C)\). The Hasse constants are used to describe the lattice of subloops of the smallest non-associative simple Moufang loop (which has 120 elements). The diagram of this lattice is presented.