On atoms in lattices of primitive classes
Jaroslav Ježek · Czech digital mathematics library · 1970
This paper is a continuation of my papers 12} and [31 on lattices & A (of all primitive classes of algebras of type A ).For the terminology see C33 .We shall be concerned with atoms in £" .It is well-known (see HI) that every £. is atomic.In § 1, Theorem 1, a complete answer to the following question (Gratzer'a problem 33 in £13 is given: find the number of atoms in i£ f for all types A .For any complete atomic lattice L we can define, in a natural way, an element of L : the supremum of the set of all atoms of L. If L *» «C 4 , then every element of L determines a primitive class of algebras of type A and we may ask to describe the primitive class determined by the supremum of atoms* The description depends on whether A contains or does not contain at least binary operations* The description is found in Theorems 2 and 3* For the terminology and notation see § 1 of 133* As in 133, we fix an infinitely countable set X and for each type A an absolutely free algebra W s of type A .I-? A is an algebra of type A m (m^^i € %