Characterizations of Hamiltonian algebras
Characterizing binary discriminator algebras
The concept of the (dual) binary discriminator was introduced by R. Halas, I. G. Rosenberg and the author in 1999. We study finite algebras having the (dual) discriminator as a term function. In particular, a simple characterization is obtained for such algebras with a majority term function.
Classification of Maps by Their Membership in Maximal Clones That Contain Minimum and Complement
Clausal relations and C-clones
We introduce a special set of relations called clausal relations. We study a Galois connection Pol-CInv between the set of all finitary operations on a finite set D and the set of clausal relations, which is a restricted version of the Galois connection Pol-Inv. We define C-clones as the Galois closed sets of operations with respect to Pol-CInv and describe the lattice of all C-clones for the Boolean case D = {0,1}. Finally we prove certain results about C-clones over a larger set.
Clone automorphisms and hypersubstitutions.
Clone compatible identities and clone extensions of algebras
Clones closed with respect to permutation groups or transformation semigroups.
Clones, coclones and coconnected spaces.
Clones in topology and algebra.
Clones on regular cardinals
We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg’s theorem: there are maximal (= “precomplete”) clones on a set of size λ. The clones we construct do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem from pcf theory we show that for cardinals λ (in particular, for all successors of...
Commutation of operations and its relationship with Menger and Mann superpositions
The article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.
Completeness criteria and invariants for operation and transformation algebras.
Completeness, functional completeness and relative completeness.
Complexity of terms, composition, and hypersubstitution.
Computability potential scales of -element algebras with restrictions on arity.
Congruence classes in regular varieties.
Congruence preserving operations on the ring
We investigate the interval in the lattice of clones on the ring between the clone of polynomial operations and the clone of congruence preserving operations. All clones in this interval are known and described by means of generators. In this paper, we characterize each of these clones by the property of preserving a small set of relations. These relations turn out to be in a close connection to commutators.
Congruences on the subclones of the rank 3 Burle clone containing no creative functions.
Conjugated algebras
We generalize the correspondence between basic algebras and lattices with section antitone involutions to a more general case where no lattice properties are assumed. These algebras are called conjugated if this correspondence is one-to-one. We get conditions for the conjugary of such algebras and introduce the induced relation. Necessary and sufficient conditions are given to indicated when the induced relation is a quasiorder which has “nice properties", e.g. the unary operations are antitone...
Convex sets in algebras