Some universal properties of the category of clones. (Q1771917)

The lattice \(\mathcal L\) of interpretability types of varieties of universal algebras was studied by \textit{O.\ C.\ García} and \textit{W.\ Taylor} [The lattice of interpretability types of varieties. Mem. Am. Math. Soc. 305 (1984; Zbl 0559.08003)] and many open problems were formulated there. In the present paper the authors solve one of the problems, namely the question: what are the possible cardinalities of antichains of \(\mathcal L\)? It is proved: (1) that every cardinal number can be the cardinality of an antichain in \(\mathcal L\), and (2) that the existence of a proper-class antichain in \(\mathcal L\) is equivalent to the negation of Vopěnka's set-theoretical principle.