ИСТИНА

In 1970 Ivo Rosenberg classified all maximal clones on finite sets, and one of the families in the classification consists of maximal clones defined by central relations, where a relation is called central if 1) it is totally reflexive, i.e., it contains all tuples where not all elements are different; 2) it is totally symmetric, i.e., any permutation of variables gives the same relation; 3) there exists a central element $c$, i.e. it contains all tuples with $c$. The center is the set of all elements $c$ with this property. We can generalize the notion of a center to any relation, where the center is the set of all elements $c$ such that the relation contains all tuples starting with $c$. Earlier we proved that a center of an invariant subdirect relation (in Taylor algebra without binary absorption) absorbs the universe. Since absorption does not imply a center in general, the connection of a center and absorption remained unclear. In the talk we clarify this question and show that a center is very similar to a ternary absorbing set. Precisely, we prove that a center (in Taylor algebra without binary absorption) implies a ternary absorption, and a ternary absorbing set in a minimal Taylor algebra is a center. Additionally, we consider some implications of this result to clone theory.