Algebraic functions on p-rings
Algebraic lattices are complete sublattices of the clone lattice over an infinite set
The clone lattice Cl(X) over an infinite set X is a complete algebraic lattice with compact elements. We show that every algebraic lattice with at most compact elements is a complete sublattice of Cl(X).
Algebraische Hüllenoperatoren, bei denen jede endliche Menge durch die Menge ihrer isolierten Elemente erzeugt wird
Algebras of multiplace functions.
All completely regular elements in
In Universal Algebra, identities are used to classify algebras into collections, called varieties and hyperidentities are use to classify varieties into collections, called hypervarities. The concept of a hypersubstitution is a tool to study hyperidentities and hypervarieties. Generalized hypersubstitutions and strong identities generalize the concepts of a hypersubstitution and of a hyperidentity, respectively. The set of all generalized hypersubstitutions forms a monoid. In...
Almost associative operations generating a minimal clone
Characterizations of 'almost associative' binary operations generating a minimal clone are given for two interpretations of the term 'almost associative'. One of them uses the associative spectrum, the other one uses the index of nonassociativity to measure how far an operation is from being associative.
Approximation durch Polynomfunktionen auf universellen Algebren.
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.