Some examples of primitive lattices
Jaroslav Ježek, Václav Slavík · Czech digital mathematics library · 1973
Igosin [1] investigates characterizable primitive classes of lattices, i.e. primitive classes K such that any lattice belongs to K iff it does not contain a sublattice belonging to a given set of finite lattices.In the present paper we shall be concerned with primitive classes characterizable by means of a single lattice.We give some examples of them and prove that there are infinitely many such classes.Given a lattice L, denote by K(L) the class of all lattices that contain no sublattice isomorphic to L. We call L primitive if the class K(L) is primitive.THEOREM 1.Let L be an arbitrary lattice.The following holds: (1) The class K(L) is closed with respect to sublattices and isomorphic lattices.(2) If L is subdirectly irreducible, then K(L) is closed with respect to direct products.(3) If L is finite and K(L) is closed with respect to direct products, then L is subdirectly irreducible.Proof.(1) is trivial.Let us prove (2).Suppose that there exists an isomorphism j of L onto a sublattice of the direct product X At of a family of lattices At e K(L).teT