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Displaying similar documents to “Locally finite M-solid varieties of semigroups”

k-Normalization and (k+1)-level inflation of varieties

Valerie Cheng, Shelly Wismath (2008)

Discussiones Mathematicae - General Algebra and Applications

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Let τ be a type of algebras. A common measurement of the complexity of terms of type τ is the depth of a term. For k ≥ 1, an identity s ≈ t of type τ is said to be k-normal (with respect to this depth complexity measurement) if either s = t or both s and t have depth ≥ k. A variety is called k-normal if all its identities are k-normal. Taking k = 1 with respect to the usual depth valuation of terms gives the well-known property of normality of identities or varieties. For any variety...

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Klaus Denecke, Jörg Koppitz, Nittiya Pabhapote (2008)

Discussiones Mathematicae - General Algebra and Applications

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A regular hypersubstitution is a mapping which takes every n i -ary operation symbol to an n i -ary term. A variety is called regular-solid if it contains all algebras derived by regular hypersubstitutions. We determine the greatest regular-solid variety of semigroups. This result will be used to give a new proof for the equational description of the greatest solid variety of semigroups. We show that every variety of semigroups which is finitely based by hyperidentities is also finitely based...

The dimension of a variety

Ewa Graczyńska, Dietmar Schweigert (2007)

Discussiones Mathematicae - General Algebra and Applications

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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...

Complexity of hypersubstitutions and lattices of varieties

Thawhat Changphas, Klaus Denecke (2003)

Discussiones Mathematicae - General Algebra and Applications

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Hypersubstitutions are mappings which map operation symbols to terms. The set of all hypersubstitutions of a given type forms a monoid with respect to the composition of operations. Together with a second binary operation, to be written as addition, the set of all hypersubstitutions of a given type forms a left-seminearring. Monoids and left-seminearrings of hypersubstitutions can be used to describe complete sublattices of the lattice of all varieties of algebras of a given type. The...

A note on normal varieties of monounary algebras

Ivan Chajda, Helmut Länger (2002)

Czechoslovak Mathematical Journal

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A variety is called normal if no laws of the form s = t are valid in it where s is a variable and t is not a variable. Let L denote the lattice of all varieties of monounary algebras ( A , f ) and let V be a non-trivial non-normal element of L . Then V is of the form M o d ( f n ( x ) = x ) with some n > 0 . It is shown that the smallest normal variety containing V is contained in H S C ( M o d ( f m n ( x ) = x ) ) for every m > 1 where C denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of L consisting of all normal...

T-Varieties and Clones of T-terms

Klaus Denecke, Prakit Jampachon (2005)

Discussiones Mathematicae - General Algebra and Applications

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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...

On universality of semigroup varieties

Marie Demlová, Václav Koubek (2006)

Archivum Mathematicum

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A category K is called α -determined if every set of non-isomorphic K -objects such that their endomorphism monoids are isomorphic has a cardinality less than α . A quasivariety Q is called Q -universal if the lattice of all subquasivarieties of any quasivariety of finite type is a homomorphic image of a sublattice of the lattice of all subquasivarieties of Q . We say that a variety V is var-relatively alg-universal if there exists a proper subvariety W of V such that homomorphisms of V whose...

On the dimension of secant varieties

Luca Chiantini, Ciro Ciliberto (2010)

Journal of the European Mathematical Society

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In this paper we generalize Zak’s theorems on tangencies and on linear normality as well as Zak’s definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumptions on X , and we classify varieties for which the bound is attained.