Some universal properties of the category of clones
by
A. Barkhudaryan
Charles University, Prague
Coauthors: V. Trnková (Charles University, Prague)

In [1], O. C. García and W. Taylor formulated the following
Problem: Is the breadth of the lattice L of interpretability types of varieties uncountable? Is there an antichain which is a proper class?

The paper [5] solves the above stated problem. It comes out that

(i) any cardinal number is the cardinality of an antichain in L and
(ii) the existence of a proper class antichain in L is equivalent to the negation of the set-theoretical Vopenka's principle.

(Vopenka's principle can be understood as a statements about large cardinals; it implies the existence of a measurable cardinal and its consistency follows from the existence of a huge cardinal. See [2] for details.)

The paper actually includes two constructions (which will be outlined in the contribution). The first one gives in some sense a minimal solution of the problem. The second one gives stronger results about the category Clone of all clones and their homomorphisms at the cost of being more complicated. We mention the main consequences of the second construction:

(i) there exist arbitrarily large rigid sets of clones (a set S of objects of a category K is called rigid if whenever A, B in S and f:A --> B is a K-morphism, then A=B and f is the identity);
(ii) the existence of a rigid proper class of clones is equivalent to the negation of Vopenka's principle;
(iii) each group can be represented as the monoid of endomorphisms of some clone.

It should be also mentioned that the clones this construction gives are simple, meaning they have no non-trivial congruences.

The question whether Clone is algebraically universal in the sense of [4] remains open.

References

1.  O. C. García, W. Taylor, The lattice of interpretability types of varieties , Mem. Amer. Math. Soc. 50 (1984), no. 305.

2.  T. Jech, Set Theory , Academic Press, New York, 1978.

3.  R. N. McKenzie, G. F. McNulty, W. F. Taylor, Algebras, Lattices, Varieties , Volume 1, Monterey, California, 1978.

4.  A. Pultr, V. Trnková, Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories , North-Holland, Amsterdam, 1980.

5.  V. Trnková, A. Barkhudaryan, Some universal properties of the category of clones , preprint.

Date received: June 1, 1999