TY  - JOUR
A1  - Denecke, Klaus-Dieter
T1  - Partial clones
JF  - Asian-European journal of mathematics : AEJM
N2  - 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.
KW  - Operation
KW  - term
KW  - formula
KW  - superposition of operations
KW  - terms and
KW  - formulas
KW  - linear term
KW  - linear formula
KW  - linear tree language
KW  - clone
KW  - partial clone
KW  - linear hypersubstitution
KW  - dht-symmetric category
KW  - partial
KW  - theory
Y1  - 2020
U6  - https://doi.org/10.1142/S1793557120501612
SN  - 1793-5571
SN  - 1793-7183
VL  - 13
IS  - 8
PB  - World Scientific
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  -