TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Hybrid identities and hybrid equational logic Y1 - 1995 ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Clones and hyperidentities Y1 - 1995 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Koppitz, Jörg A1 - Štrakov, Slavčo T1 - Multi-hypersubstitutions and colored solid varieties JF - International journal of algebra and computation N2 - Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently-colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multihypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice. KW - coloration of terms KW - multi-hypersubstitutions KW - colored solid varieties Y1 - 2006 U6 - https://doi.org/10.1142/S0218196706003189 SN - 0218-1967 VL - 16 IS - 4 SP - 797 EP - 815 PB - World Scient. Publ. CY - Singapore ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - The partial clone of linear formulas JF - Siberian mathematical journal N2 - A term t is linear if no variable occurs more than once in t. An identity s ≈ t is said to be linear if s and t are linear terms. Identities are particular formulas. As for terms superposition operations can be defined for formulas too. We define the arbitrary linear formulas and seek for a condition for the set of all linear formulas to be closed under superposition. This will be used to define the partial superposition operations on the set of linear formulas and a partial many-sorted algebra Formclonelin(τ, τ′). This algebra has similar properties with the partial many-sorted clone of all linear terms. We extend the concept of a hypersubstitution of type τ to the linear hypersubstitutions of type (τ, τ′) for algebraic systems. The extensions of linear hypersubstitutions of type (τ, τ′) send linear formulas to linear formulas, presenting weak endomorphisms of Formclonelin(τ, τ′). KW - term KW - formula KW - superposition KW - linear term KW - linear formula KW - clone KW - partial clone KW - linear hypersubstitution Y1 - 2019 U6 - https://doi.org/10.1134/S0037446619040037 SN - 0037-4466 SN - 1573-9260 VL - 60 IS - 4 SP - 572 EP - 584 PB - Pleiades Publ. CY - New York ER - TY - JOUR A1 - Arworn, Srichan A1 - Denecke, Klaus-Dieter T1 - Groupoids of hypersubstitutions and G-solid varieties Y1 - 1997 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Poomsa-ard, T. T1 - Hyperidentities in graph algebras Y1 - 1997 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Wismath, Shelly T1 - The monoid of hypersubstitutions of type (2) Y1 - 1997 ER - TY - JOUR A1 - Arworn, Srichan A1 - Denecke, Klaus-Dieter T1 - A new methods to study subvariety lattices of semigroup varieties Y1 - 1997 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Lüders, Otfried T1 - Category equivalences of clones Y1 - 1995 ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - The entropy sequence of unary logical functions Y1 - 1996 ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Clones and Hyperidentities Y1 - 1997 ER - TY - BOOK A1 - Denecke, Klaus-Dieter A1 - Todorov, Kalco T1 - Allgemeine Algebra und Anwendungen Y1 - 1996 PB - Shaker CY - Aachen ER - TY - BOOK A1 - Denecke, Klaus-Dieter T1 - Clones and hyperidentities T3 - Preprint / Universität Potsdam, Institut für Mathematik Y1 - 1996 VL - 1996, 14 PB - Univ. CY - Potsdam ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - The partial clone of linear terms JF - Siberian Mathematical Journal N2 - Generalizing a linear expression over a vector space, we call a term of an arbitrary type tau linear if its every variable occurs only once. Instead of the usual superposition of terms and of the total many-sorted clone of all terms in the case of linear terms, we define the partial many-sorted superposition operation and the partial many-sorted clone that satisfies the superassociative law as weak identity. The extensions of linear hypersubstitutions are weak endomorphisms of this partial clone. For a variety V of one-sorted total algebras of type tau, we define the partial many-sorted linear clone of V as the partial quotient algebra of the partial many-sorted clone of all linear terms by the set of all linear identities of V. We prove then that weak identities of this clone correspond to linear hyperidentities of V. KW - linear term KW - clone KW - partial clone KW - linear hypersubstitution KW - linear identity KW - linear hyperidentity Y1 - 2016 U6 - https://doi.org/10.1134/S0037446616040030 SN - 0037-4466 SN - 1573-9260 VL - 57 SP - 589 EP - 598 PB - Pleiades Publ. CY - New York ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Wismath, Shelly T1 - The dimension of a variety and the kernel of a hypersubstitution N2 - The dimension of a variety V of algebras of a given type was introduced by E. Graczynska and D. Schweigert in [7] as the cardinality of the set of all derived varieties of V which are properly contained in V. In this paper, we characterize all solid varieties of dimensions 0, 1, and 2; prove that the dimension of a variety of finite type is at most N-0; give an example of a variety which has infinite dimension; and show that for every n is an element of N there is a variety with dimension n. Finally, we show that the dimension of a variety is related to the concept of the semantical kernel of a hypersubstitution and apply this connection to calculate the dimension of the class of all algebras of type tau = (n). Y1 - 2009 UR - http://www.worldscinet.com/ijac/ U6 - https://doi.org/10.1142/S0218196709005342 SN - 0218-1967 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Saengsura, Kittisak T1 - Separation of clones of cooperations by cohyperidentities N2 - An n-ary cooperation is a mapping from a nonempty set A to the nth copower of A. A clone of cooperations is a set of cooperations which is closed under superposition and contains all injections. Coalgebras are pairs consisting of a set and a set of cooperations defined on this set. We define terms for coalgebras, coidentities and cohyperidentities. These concepts will be applied to give a new solution of the completeness problem for clones of cooperations defined on a two-element set and to separate clones of cooperations by coidentities. Y1 - 2009 UR - http://www.sciencedirect.com/science/journal/0012365X U6 - https://doi.org/10.1016/j.disc.2008.01.043 SN - 0012-365X ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Congruences on maximal partial clones and strong regular varieties generated by preprimal partial algebras II Y1 - 1991 ER - TY - BOOK A1 - Denecke, Klaus-Dieter A1 - Wismath, Shelly L. T1 - Universal Algebra and Coalgebra Y1 - 2009 SN - 978-981-283745-5 PB - World Scientific Publ. Co CY - Singapore ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Strong regular varieties of partial algebras I Y1 - 1991 ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Minimal algebras and category equivalences Y1 - 1991 ER - TY - BOOK ED - Denecke, Klaus-Dieter T1 - General algebra and applications T3 - Research and exposition in mathematics Y1 - 1993 SN - 3-88538-220-2 VL - 20 PB - Heldermann CY - Berlin ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Koppitz, Jörg T1 - A characterization of M-solid varieties of semigroups Y1 - 1999 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Jampachon, Prakit T1 - Regular-solid varieties of commutative and idempotent groupoids Y1 - 1999 ER - TY - JOUR A1 - Wismath, Shelly A1 - Koppitz, Jörg A1 - Denecke, Klaus-Dieter T1 - Maps between M-solid varieties of emigroups Y1 - 1997 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Koppitz, Jörg A1 - Marszalek, R. T1 - Derived varieties and derived equational theories Y1 - 1998 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Koppitz, Jörg T1 - M-solid monoids of hypersubstitutions of type 2 Y1 - 1998 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Marszalek, R. T1 - Binary relations on monoids of hypersubstitutions Y1 - 1997 ER - TY - BOOK ED - Denecke, Klaus-Dieter T1 - General algebra and applications in discrete mathematics : proceedings of "Conference on General Algebra and Discrete Mathematics" T3 - Berichte aus der Mathematik Y1 - 1997 SN - 3-8265-2431-4 PB - Shaker CY - Aachen ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Tame congruence theory Y1 - 1998 ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Hyperequational Theorie Y1 - 1998 ER - TY - BOOK A1 - Denecke, Klaus-Dieter T1 - Hyperequational theory T3 - Preprint / Universität Potsdam, Institut für Mathematik Y1 - 1997 VL - 1997, 29 PB - Univ. CY - Potsdam ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Welke, Dirk T1 - Solid varieties of partial algebras Y1 - 1997 SN - 5-7782-0175-3 (vol ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Lau, Dietlinde A1 - Pöschel, Reinhard A1 - Schweigert, D. T1 - Free clones and solid varieties Y1 - 1995 ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - On the characterization of primal partial algebras by strong regular hyperidentities Y1 - 1994 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Halkowska, Katarzcyna T1 - P-compatible hybrid-identities and hyperidentities Y1 - 1994 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Malcev, I. A. T1 - Separation of clones by means of hyperidentities Y1 - 1994 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Wismath, Shelly T1 - Solid varieties of semigroups Y1 - 1994 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Koppitz, Jörg T1 - Hyperassociative semigroups Y1 - 1994 ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Pre-solid varieties Y1 - 1994 ER - TY - BOOK A1 - Denecke, Klaus-Dieter A1 - Todorov, Kalco T1 - Algebraische Grundlagen der Arithmetik T3 - Berliner Studienreihe zur Mathematik Y1 - 1994 VL - 4 PB - Heldermann CY - Berlin ER - TY - JOUR A1 - Lekkoksung, Nareupanat A1 - Denecke, Klaus-Dieter T1 - The partial clone of linear tree languages JF - Siberian mathematical journal N2 - A term, also called a tree, is said to be linear, if each variable occurs in the term only once. The linear terms and sets of linear terms, the so-called linear tree languages, play some role in automata theory and in the theory of formal languages in connection with recognizability. We define a partial superposition operation on sets of linear trees of a given type and study the properties of some many-sorted partial clones that have sets of linear trees as elements and partial superposition operations as fundamental operations. The endomorphisms of those algebras correspond to nondeterministic linear hypersubstitutions. KW - linear term KW - linear tree language KW - clone KW - partial clone KW - linear hypersubstitution KW - nondeterministic linear hypersubstitution Y1 - 2019 U6 - https://doi.org/10.1134/S0037446619030121 SN - 0037-4466 SN - 1573-9260 VL - 60 IS - 3 SP - 497 EP - 507 PB - Pleiades Publ. CY - New York ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Pibaljommee, Bundit T1 - Clones of implicit operations N2 - There is a close connection between a variety and its clone. The clone of a variety is a multibased algebra, where the different universes are the sets of n-ary terms over this variety for every natural number n and where the operations describe the superposition of terms of different arities. All projections are added as nullary operations. Subvarieties correspond to homomorphic images of clones. Subclones can be described by reducts of varieties, isomorphic clones by equivalent varieties. Clone identities correspond to hyperidentities and varieties of clones to hypervarieties. Pseudovarieties are classes of finite algebras which are closed under taking of subalgebras, homomorphic images and finite direct products. Pseudovarieties are important in the theories of finite state automata, rational languages, finite semigroups and their connections. In a very natural way, there arises the question for the clone of a pseudovariety. In the present paper, we will describe this algebraic structure Y1 - 2005 SN - 0002-5240 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Radelecki, S. A1 - Ratanaprasert, C. T1 - On constantive simple and order-primal algebras N2 - A finite algebra A = (A; F-A) is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order <= on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; <=). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case F-A cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras Y1 - 2005 ER - TY - JOUR A1 - Changphas, Thawhat A1 - Denecke, Klaus-Dieter T1 - Green's relation R on the monoid of clone endomorphisms N2 - A hypersubstitution is a map which takes n-ary operation symbols to n-ary terms. Any such map can be uniquely extended to a map defined on the set W-tau(X) of all terms of type tau, and any two such extensions can be composed in a natural way. Thus, the set Hyp(tau) of all hypersubstitutions of type tau forms a monoid. In this paper, we characterize Green's relation R on the monoid Hyp(tau) for the type tau = (n, n). In this case, the monoid of all hypersubstitutions is isomorphic with the monoid of all Clone endomorphisms. The results can be applied to mutually derived varieties Y1 - 2005 SN - 1005-3867 ER - TY - JOUR A1 - Changphas, Thawhat A1 - Denecke, Klaus-Dieter T1 - Green's Relations on the Seminearring of Full Hypersubstitutions of Type (n) Y1 - 2003 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Pibaljommee, Bundit T1 - Locally finite M-solid Varieties of Semigroups Y1 - 2003 ER - TY - JOUR A1 - Changphas, Thawhat A1 - Denecke, Klaus-Dieter T1 - Complexity of Hypersubstitutions and Lattices of Varieties Y1 - 2003 ER - TY - THES A1 - Supaporn, Worakrit T1 - Categorical equivalence of clones Y1 - 2014 ER - TY - BOOK A1 - Denecke, Klaus-Dieter T1 - Algebra und Diskrete Mathematik für Informatiker N2 - Algebra und Diskrete Mathematik gehören zu den wesentlichen Grundlagen der Informatik. Sie sind unverzichtbare Werkzeuge eines jeden Informatikers und spielen daher auch im Studium eine zentrale Rolle. Das Lehrbuch vermittelt anschaulich und leicht nachvollziehbar die wichtigsten algebraischen Grundlagen der Informatik bis hin zur Gleichungstheorie der Universellen Algebra. Zahlreiche Übungsaufgaben und ihre Lösungen helfen dem Leser, den Stoff zu verstehen Y1 - 2003 PB - Teubner CY - Stuttgart u.a. ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Jampachon, Prakit T1 - N-solid Varieties and free Menger Algebras of rRnk n Y1 - 2003 ER -