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On the one hand we determine all conservative self-commuting binary operations on any set, on the other hand we obtain a full list of selfcommuting operations of arbitrary arity on the 2-element set. [ABSTRACT FROM AUTHOR]

A set of hyperoperations on a finite set A is upward saturated if it contains all super-hyperoperations of each hyperoperation belonging to the set. The set of all upward saturated hyperclones, ordered by set inclusion, is an algebraic lattice.We describe a class of maximal hyperclones determined by bounded partial orders. Hyperclones from this class are upward saturated. [ABSTRACT FROM AUTHOR]

Incompletely specified operations on a finite set are operations with values specified only for some elements of the domain. The set of all such mappings, together with naturally introduced fundamental operations, forms the algebra of incompletely specified operations. On a two element set, it is isomorphic to the full algebra of hyperoperations. On a set with at least three elements, there is no suitable homomorphism to algebras of total, partial or hyperoperations. An incompletely specified operation on a set induces an operation on the set extended with one additional element. We consider the full algebra of those extended incompletely specified operations. In general, sets of extended incompletely specified operations that preserve a given relation are not closed under fundamental operations of the algebra. However, certain classes of relations possess this property and two of them are presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we present three classes of maximal hyperclones. They are determined by three classes of Rosenberg's relations: nontrivial equivalence relations, central relations and h-regular relations. [ABSTRACT FROM AUTHOR]

For an algebra A on a universe A the set of endomorphisms of A forms a monoid of selfmaps on A. We call such monoids centralizing monoids. In this paper we study the first interesting case of a 3-element set A. In the lattice of inclusion-ordered centralizing monoids we find three maximal centralizing monoids. This knowledge suggests that many large monoids are not centralizing monoids. [ABSTRACT FROM AUTHOR]