Algebraic Lattices Are Complete Sublattices of the Clone Lattice over an Infinite Set

The clone lattice Cl(X) over an infinite set X is a complete algebraic lattice with 2 compact elements. We show that every algebraic lattice with at most 2 compact elements is a complete sublattice of Cl(X). 1. How complicated is the clone lattice? Fix a base set X and denote for all n ≥ 1 the set of all n-ary operations on X by O(n). Then O = ⋃ n≥1 O (n) is the set of all functions on X which have finite arity. A set of finitary functions C ⊆ O is called a clone iff it is closed under composition and contains all projections, i.e. for all 1 ≤ i ≤ n the function π i satisfying π n i (x1, . . . , xn) = xi. The set of all clones over X forms a complete algebraic lattice Cl(X) with respect to inclusion. This lattice is countably infinite and completely known if |X| = 2 by a result of Post’s [Pos41]; however, describing the clone lattice completely for larger X is believed impossible. Several known results suggest this: To begin with, Cl(X) is large; it is of size continuum if X is finite and has at least three elements, and |Cl(X)| = 22 |X| if X is infinite. Then, the clone lattice does not satisfy any non-trivial lattice identity if |X| ≥ 3 [Bul93]; it does not satisfy any quasiidentity if |X| ≥ 4 [Bul94]. Also, if |X| ≥ 4, then every countable product of finite lattices is a sublattice of Cl(X) [Bul94]. As for an example on infinite X, every completely distributive lattice having not more than 2|X| compact elements is a subinterval of a monoidal interval of Cl(X) [Pin] (a monoidal interval being an interval of clones which have the same unary functions). We are interested in which lattices can be embedded into the clone lattice over an infinite set. Assume henceforth X to be infinite. The compact elements of Cl(X) are easily seen to be exactly the clones which are generated by a finite number of functions. Since |O| = 2|X|, this implies that Cl(X) has at most 2|X| compact elements, and it is readily verified that there really exist 2|X| compact elements. We are going to prove that Cl(X) is in some sense the most complicated algebraic lattice with this property. 1991 Mathematics Subject Classification. Primary 08A40; secondary 08A05.