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  - Wismath, Shelly
T1  - Solid varieties of semigroups
Y1  - 1994
ER  - 
TY  - JOUR
A1  - Denecke, Klaus-Dieter
A1  - Wismath, Shelly
T1  - The monoid of hypersubstitutions of type (2)
Y1  - 1997
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  - BOOK
A1  - Denecke, Klaus-Dieter
A1  - Wismath, Shelly
T1  - Hyperidentities and clones
T3  - Algebra, logic and aplications
N2  - The theory of hyperidentities generalises the equational theory of universal algebras and is applicable in several fields of science, especially in computer sciences. This book presents the theory of hyperidentities and its relation to clone identities. The basic concept of hypersubstitution is used to introduce the monoid of hypersubstitutions, hyperidentities, M-hyperidentities, solid and M-solid varieties. This work integrates into a coherent framework many results scattered throughout the literature over the last eighteen years. In addition, the book contains some applications of hyperidentities to the functional completenes problem in multiple-valued logic. The general theory is also extended to partial algberas. The last chapter contains a list of exercises and open problems with suggestions of future work in this area of research.
Y1  - 2000
SN  - 90-5699-235-X
VL  - 14
PB  - Gordon & Breach
CY  - Amsterdam
ER  - 
TY  - JOUR
A1  - Denecke, Klaus-Dieter
A1  - Koppitz, Jörg
A1  - Wismath, Shelly
T1  - The semantical hyperunification problem
Y1  - 2001
ER  - 
TY  - BOOK
A1  - Denecke, Klaus-Dieter
A1  - Wismath, Shelly
T1  - Universal algebra and applications in theoretical computer science
Y1  - 2002
SN  - 1-584-88254-9
PB  - Chapman & Hall/CRC
CY  - Boca Raton
ER  - 
TY  - JOUR
A1  - Denecke, Klaus-Dieter
A1  - Wismath, Shelly
T1  - M-solidity testing systems
Y1  - 2002
ER  - 
TY  - JOUR
A1  - Denecke, Klaus-Dieter
A1  - Koppitz, Jörg
A1  - Wismath, Shelly
T1  - Solid Varietie of Arbitrary Type
Y1  - 2002
ER  - 
TY  - JOUR
A1  - Denecke, Klaus-Dieter
A1  - Wismath, Shelly
T1  - Valuations of Terms
N2  - 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
Y1  - 2003
ER  -