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The dimension of a variety

Ewa Graczyńska, Dietmar Schweigert (2007)

Discussiones Mathematicae - General Algebra and Applications

Derived varieties were invented by P. Cohn in [4]. Derived varieties of a given type were invented by the authors in [10]. In the paper we deal with the derived variety V σ of a given variety, by a fixed hypersubstitution σ. We introduce the notion of the dimension of a variety as the cardinality κ of the set of all proper derived varieties of V included in V. We examine dimensions of some varieties in the lattice of all varieties of a given type τ. Dimensions of varieties of lattices and all subvarieties...

The weak hereditary class of a variety

Wiktor Bartol, Francesc Rosselló (2006)

Czechoslovak Mathematical Journal

We study the weak hereditary class S w ( 𝒦 ) of all weak subalgebras of algebras in a total variety 𝒦 . We establish an algebraic characterization, in the sense of Birkhoff’s HSP theorem, and a syntactical characterization of these classes. We also consider the problem of when such a weak hereditary class is weak equational.

T-Varieties and Clones of T-terms

Klaus Denecke, Prakit Jampachon (2005)

Discussiones Mathematicae - General Algebra and Applications

The aim of this paper is to describe how varieties of algebras of type τ can be classified by using the form of the terms which build the (defining) identities of the variety. There are several possibilities to do so. In [3], [19], [15] normal identities were considered, i.e. identities which have the form x ≈ x or s ≈ t, where s and t contain at least one operation symbol. This was generalized in [14] to k-normal identities and in [4] to P-compatible identities. More generally, we select a subset...

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