On minimal and maximal clones

  László Szabó

 A composition closed set of finitary operations on a fixed universe containing all projections is a clone. For example the set of all projections and the set O of all operations on are clones. The clones, ordered by inclusion, form an algebraic lattice with least element and greatest element . For , is the well-known countable Post lattice [], but already for there are clones. For finite has finitely many coatoms, called maximal clones , and they are fully known ([],[]). On the other hand has finitely many atoms, called minimal clones, and are fully known only for ([], []). It is also known (see e.g. []) that the meet of all maximal clones is , and the join of all minimal clones is .

 The aim of the present paper is to show that in general there are three maximal clones with meet and there are three minimal clones with join ; moreover, for a prime element universe, two maximal clones, resp., two minimal clones have the above properties.

 Gyenizse Pal 1996. Szeptember 4. Szerda 13:46:35 MET DST