TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Wismath, Shelly T1 - A characterization of k-normal varieties N2 - Let v be a valuation of terms of type tau, assigning to each term t of type tau a value v(t) greater than or equal to 0. Let k greater than or equal to 1 be a natural number. An identity s approximate to t of type tau is called k- normal if either s = t or both s and t have value greater than or equal to k, and otherwise is called non-k-normal. A variety V of type tau is said to be k-normal if all its identities are k-normal, and non-k-normal otherwise. In the latter case, there is a unique smallest k-normal variety N-k(A) (V) to contain V , called the k-normalization of V. Inthe case k = 1, for the usual depth valuation of terms, these notions coincide with the well-known concepts of normal identity, normal variety, and normalization of a variety. I. Chajda has characterized the normalization of a variety by means of choice algebras. In this paper we generalize his results to a characterization of the k-normalization of a variety, using k-choice algebras. We also introduce the concept of a k-inflation algebra, and for the case that v is the usual depth valuation of terms, we prove that a variety V is k-normal iff it is closed under the formation of k- inflations, and that the k-normalization of V consists precisely of all homomorphic images of k-inflations of algebras in V Y1 - 2004 SN - 0002-5240 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Jampachon, Prakit A1 - Wismath, Shelly T1 - Clones of n-ary algebras Y1 - 2003 ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Wismath, Shelly T1 - Complexity of Terms, Composition and Hypersubstitution Y1 - 2003 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 - Wismath, Shelly T1 - M-solidity testing systems Y1 - 2002 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 - Wismath, Shelly T1 - Solid Varietie of Arbitrary Type Y1 - 2002 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 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 - The monoid of hypersubstitutions of type (2) Y1 - 1997 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 - Valuations and Hypersubstitutions Y1 - 2003 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 -