@article{DeneckeLeeratanavalee1999,
  author    = {Denecke, Klaus-Dieter and Leeratanavalee, Sorasak},
  title     = {Weak hypersubstitutions and weakly derived algebras},
  year      = {1999},
  language  = {en}
}
@article{DeneckeWismath2003,
  author    = {Denecke, Klaus-Dieter and Wismath, Shelly},
  title     = {Valuations of Terms},
  year      = {2003},
  abstract  = {Let tau be a type of algebras. There are several commonly used measurements of the complexity of terms of type tau, including the depth or height of a term and the number of variable symbols appearing in a term. In this paper we formalize these various measurements, by defining a complexity or valuation mapping on terms. A valuation of terms is thus a mapping from the absolutely free term algebra of type tau into another algebra of the same type on which an order relation is defined. We develop the interconnections between such term valuations 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; valuations of terms offer a new method to study complete sublattices of this lattice},
  language  = {en}
}
@article{DeneckeWismath2003,
  author    = {Denecke, Klaus-Dieter and Wismath, Shelly},
  title     = {Valuations and Hypersubstitutions},
  year      = {2003},
  language  = {en}
}
@book{DeneckeWismath2009,
  author    = {Denecke, Klaus-Dieter and Wismath, Shelly L.},
  title     = {Universal Algebra and Coalgebra},
  publisher = {World Scientific Publ. Co},
  address   = {Singapore},
  isbn      = {978-981-283745-5},
  pages     = {278 S.},
  year      = {2009},
  language  = {en}
}
@book{DeneckeWismath2002,
  author    = {Denecke, Klaus-Dieter and Wismath, Shelly},
  title     = {Universal algebra and applications in theoretical computer science},
  publisher = {Chapman \& Hall/CRC},
  address   = {Boca Raton},
  isbn      = {1-584-88254-9},
  pages     = {383 S.},
  year      = {2002},
  language  = {en}
}
@article{DeneckePabhapote2001,
  author    = {Denecke, Klaus-Dieter and Pabhapote, Nittiya},
  title     = {Tree-recognizers and tree-hyperrecognizers},
  year      = {2001},
  language  = {en}
}
@article{ArwornDenecke2001,
  author    = {Arworn, Srichan and Denecke, Klaus-Dieter},
  title     = {Tree Transformations defined by Hypersubstitutions},
  issn      = {1509 - 9415},
  year      = {2001},
  language  = {en}
}
@article{DeneckeLeeratanavalee2003,
  author    = {Denecke, Klaus-Dieter and Leeratanavalee, Sorasak},
  title     = {Tree transformations defined by generalized hypersubstitutions},
  year      = {2003},
  language  = {en}
}
@article{DeneckeFreiberg1998,
  author    = {Denecke, Klaus-Dieter and Freiberg, L.},
  title     = {The word problem for M-solid varieties of semigroups},
  isbn      = {981-3083-86-7},
  year      = {1998},
  language  = {en}
}
@article{DeneckeLeeratanavalee2003,
  author    = {Denecke, Klaus-Dieter and Leeratanavalee, Sorasak},
  title     = {The Semantical Kernel of a Generalized Hypersubstitution},
  year      = {2003},
  language  = {en}
}
@article{DeneckeKoppitzWismath2001,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Wismath, Shelly},
  title     = {The semantical hyperunification problem},
  year      = {2001},
  language  = {en}
}
@article{LekkoksungDenecke2019,
  author    = {Lekkoksung, Nareupanat and Denecke, Klaus-Dieter},
  title     = {The partial clone of linear tree languages},
  series = {Siberian mathematical journal},
  volume    = {60},
  journal   = {Siberian mathematical journal},
  number    = {3},
  publisher = {Pleiades Publ.},
  address   = {New York},
  issn      = {0037-4466},
  doi       = {10.1134/S0037446619030121},
  pages     = {497 -- 507},
  year      = {2019},
  abstract  = {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.},
  language  = {en}
}
@article{Denecke2016,
  author    = {Denecke, Klaus-Dieter},
  title     = {The partial clone of linear terms},
  series = {Siberian Mathematical Journal},
  volume    = {57},
  journal   = {Siberian Mathematical Journal},
  publisher = {Pleiades Publ.},
  address   = {New York},
  issn      = {0037-4466},
  doi       = {10.1134/S0037446616040030},
  pages     = {589 -- 598},
  year      = {2016},
  abstract  = {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.},
  language  = {en}
}
@article{Denecke2019,
  author    = {Denecke, Klaus-Dieter},
  title     = {The partial clone of linear formulas},
  series = {Siberian mathematical journal},
  volume    = {60},
  journal   = {Siberian mathematical journal},
  number    = {4},
  publisher = {Pleiades Publ.},
  address   = {New York},
  issn      = {0037-4466},
  doi       = {10.1134/S0037446619040037},
  pages     = {572 -- 584},
  year      = {2019},
  abstract  = {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′).},
  language  = {en}
}
@article{DeneckeMahdavi2000,
  author    = {Denecke, Klaus-Dieter and Mahdavi, Kazem},
  title     = {The order of normal form hypersubstitutions of type 2},
  year      = {2000},
  language  = {en}
}
@article{DeneckeWismath1997,
  author    = {Denecke, Klaus-Dieter and Wismath, Shelly},
  title     = {The monoid of hypersubstitutions of type (2)},
  year      = {1997},
  language  = {en}
}
@article{Denecke2000,
  author    = {Denecke, Klaus-Dieter},
  title     = {The galois correspondence between subvariety lattices and monoids of hypersubstitutions},
  year      = {2000},
  language  = {en}
}
@article{Denecke1996,
  author    = {Denecke, Klaus-Dieter},
  title     = {The entropy sequence of unary logical functions},
  year      = {1996},
  language  = {en}
}
@article{DeneckeWismath2009,
  author    = {Denecke, Klaus-Dieter and Wismath, Shelly},
  title     = {The dimension of a variety and the kernel of a hypersubstitution},
  issn      = {0218-1967},
  doi       = {10.1142/S0218196709005342},
  year      = {2009},
  abstract  = {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).},
  language  = {en}
}
@book{DeneckeKoppitzShtraklov2001,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Shtraklov, Slavcho},
  title     = {The Depth of a Hypersubstitution},
  year      = {2001},
  language  = {en}
}
@article{Denecke1998,
  author    = {Denecke, Klaus-Dieter},
  title     = {Tame congruence theory},
  year      = {1998},
  language  = {en}
}
@article{ArwornDeneckeKoppitz2001,
  author    = {Arworn, Srichan and Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Strongly luid and weakly unsolid varieties},
  issn      = {1346-0862},
  year      = {2001},
  language  = {en}
}
@article{Denecke1991,
  author    = {Denecke, Klaus-Dieter},
  title     = {Strong regular varieties of partial algebras I},
  year      = {1991},
  language  = {en}
}
@article{DeneckeHounnon2000,
  author    = {Denecke, Klaus-Dieter and Hounnon, Hippolyte},
  title     = {Solid varieties of semirings},
  isbn      = {981-02-4392-8},
  year      = {2000},
  language  = {en}
}
@article{DeneckeWismath1994,
  author    = {Denecke, Klaus-Dieter and Wismath, Shelly},
  title     = {Solid varieties of semigroups},
  year      = {1994},
  language  = {en}
}
@article{DeneckeWelke1997,
  author    = {Denecke, Klaus-Dieter and Welke, Dirk},
  title     = {Solid varieties of partial algebras},
  isbn      = {5-7782-0175-3 (vol},
  year      = {1997},
  language  = {en}
}
@article{Denecke2000,
  author    = {Denecke, Klaus-Dieter},
  title     = {Solid varieties of normal ID-semirings},
  isbn      = {3-8265- 7983-6},
  year      = {2000},
  language  = {en}
}
@article{DeneckeHounnon2000,
  author    = {Denecke, Klaus-Dieter and Hounnon, Hippolyte},
  title     = {Solid varieties of normal ID-semirings},
  isbn      = {3-8265- 7983-6},
  year      = {2000},
  language  = {en}
}
@article{DeneckeKoppitzWismath2002,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Wismath, Shelly},
  title     = {Solid Varietie of Arbitrary Type},
  year      = {2002},
  language  = {en}
}
@article{Denecke2000,
  author    = {Denecke, Klaus-Dieter},
  title     = {Solid polynomial varieties of semigroups which are definable by identities},
  isbn      = {3-85366-951-4},
  year      = {2000},
  language  = {en}
}
@article{DeneckeLeeratanavalee2000,
  author    = {Denecke, Klaus-Dieter and Leeratanavalee, Sorasak},
  title     = {Solid polynomial varieties of semigroups which are definable by identities},
  year      = {2000},
  language  = {en}
}
@phdthesis{ArwornDenecke1999,
  author    = {Arworn, Srichan and Denecke, Klaus-Dieter},
  title     = {Sets of hypersubstitutions and set-solid varieties},
  year      = {1999},
  language  = {en}
}
@article{DeneckeSaengsura2009,
  author    = {Denecke, Klaus-Dieter and Saengsura, Kittisak},
  title     = {Separation of clones of cooperations by cohyperidentities},
  issn      = {0012-365X},
  doi       = {10.1016/j.disc.2008.01.043},
  year      = {2009},
  abstract  = {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.},
  language  = {en}
}
@article{DeneckeMalcev1994,
  author    = {Denecke, Klaus-Dieter and Malcev, I. A.},
  title     = {Separation of clones by means of hyperidentities},
  year      = {1994},
  language  = {en}
}
@article{DeneckePlonka1995,
  author    = {Denecke, Klaus-Dieter and Plonka, J.},
  title     = {Regularization and normalization of solid varieties},
  year      = {1995},
  language  = {en}
}
@article{DeneckeJampachon1999,
  author    = {Denecke, Klaus-Dieter and Jampachon, Prakit},
  title     = {Regular-solid varieties of commutative and idempotent groupoids},
  year      = {1999},
  language  = {en}
}
@article{DeneckeKoppitz1995,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Pre-solid varieties of semigroups},
  year      = {1995},
  language  = {en}
}
@article{DeneckeKoppitz1995,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Pre-solid varieties of commutative semigroups},
  year      = {1995},
  language  = {en}
}
@article{Denecke1994,
  author    = {Denecke, Klaus-Dieter},
  title     = {Pre-solid varieties},
  year      = {1994},
  language  = {en}
}
@article{DeneckeHounnon2021,
  author    = {Denecke, Klaus-Dieter and Hounnon, Hippolyte},
  title     = {Partial Menger algebras of terms},
  series = {Asian-European journal of mathematics},
  volume    = {14},
  journal   = {Asian-European journal of mathematics},
  number    = {06},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {1793-5571},
  doi       = {10.1142/S1793557121500923},
  pages     = {14},
  year      = {2021},
  abstract  = {The superposition operation S-n,S-A, n >= 1, n is an element of N, maps to each (n + 1)-tuple of n-ary operations on a set A an n-ary operation on A and satisfies the so-called superassociative law, a generalization of the associative law. The corresponding algebraic structures are Menger algebras of rank n. A partial algebra of type (n + 1) which satisfies the superassociative law as weak identity is said to be a partial Menger algebra of rank n. As a generalization of linear terms we define r-terms as terms where each variable occurs at most r-times. It will be proved that n-ary r-terms form partial Menger algebras of rank n. In this paper, some algebraic properties of partial Menger algebras such as generating systems, homomorphic images and freeness are investigated. As generalization of hypersubstitutions and linear hypersubstitutions we consider r-hypersubstitutions.U},
  language  = {en}
}
@article{Denecke2020,
  author    = {Denecke, Klaus-Dieter},
  title     = {Partial clones},
  series = {Asian-European journal of mathematics : AEJM},
  volume    = {13},
  journal   = {Asian-European journal of mathematics : AEJM},
  number    = {8},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {1793-5571},
  doi       = {10.1142/S1793557120501612},
  pages     = {19},
  year      = {2020},
  abstract  = {A set C of operations defined on a nonempty set A is said to be a clone if C is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the n-ary operations defined on set A for all natural numbers n >= 1 and the operations are the so-called superposition operations S-m(n) for natural numbers m, n >= 1 and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set A and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.},
  language  = {en}
}
@article{Denecke2000,
  author    = {Denecke, Klaus-Dieter},
  title     = {P-compatible hypersubstitutions and M_P -solid varieties},
  year      = {2000},
  language  = {en}
}
@article{DeneckeMruczek2000,
  author    = {Denecke, Klaus-Dieter and Mruczek, Krysztyna},
  title     = {P-compatible Hypersubstitutions},
  year      = {2000},
  language  = {en}
}
@article{DeneckeHalkowska1994,
  author    = {Denecke, Klaus-Dieter and Halkowska, Katarzcyna},
  title     = {P-compatible hybrid-identities and hyperidentities},
  year      = {1994},
  language  = {en}
}
@article{Denecke1994,
  author    = {Denecke, Klaus-Dieter},
  title     = {On the characterization of primal partial algebras by strong regular hyperidentities},
  year      = {1994},
  language  = {en}
}
@article{DeneckeMalcevReschke1995,
  author    = {Denecke, Klaus-Dieter and Malcev, I. A. and Reschke, M.},
  title     = {On separation of Boolean clones by means of hyperidentities},
  year      = {1995},
  language  = {en}
}
@article{DeneckeRadeleckiRatanaprasert2005,
  author    = {Denecke, Klaus-Dieter and Radelecki, S. and Ratanaprasert, C.},
  title     = {On constantive simple and order-primal algebras},
  year      = {2005},
  abstract  = {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},
  language  = {en}
}
@article{DeneckeKoppitz1999,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Normal forms of hypersubstitutions},
  year      = {1999},
  language  = {en}
}
@article{DeneckeJampachon2003,
  author    = {Denecke, Klaus-Dieter and Jampachon, Prakit},
  title     = {N-solid Varieties and free Menger Algebras of rRnk n},
  year      = {2003},
  language  = {en}
}
@article{DeneckeKoppitzŠtrakov2006,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Štrakov, Slavčo},
  title     = {Multi-hypersubstitutions and colored solid varieties},
  series = {International journal of algebra and computation},
  volume    = {16},
  journal   = {International journal of algebra and computation},
  number    = {4},
  publisher = {World Scient. Publ.},
  address   = {Singapore},
  issn      = {0218-1967},
  doi       = {10.1142/S0218196706003189},
  pages     = {797 -- 815},
  year      = {2006},
  abstract  = {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.},
  language  = {en}
}