@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{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{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{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{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}
}