[unknown]

Klaus Denecke; Jörg Koppitz; Nittiya Pabhapote

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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...

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...

Some relations on the lattice of varieties of completely regular semigroups

Mario Petrich (2002)

Bollettino dell'Unione Matematica Italiana

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On the lattice $L (C R)$ of varieties of completely regular semigroups considered as algebras with the binary multiplication and unary inversion within maximal subgroups, we study the relations $K_{l}$ , $K$ , $K_{r}$ , $T_{l}$ , $T$ , $T_{r}$ , $C$ and $L$ . Here $K$ is the kernel relation, $T$ is the trace relation, $T_{l}$ and $T_{r}$ are the left and the right trace relations, respectively, $K_{p} = K \cap T_{p}$ for $p \in \{l, r\}$ , $C$ is the core relation and $L$ is the local relation. We give an alternative definition for each of these relations $P$ of the form $U P V \Leftrightarrow U \cap \tilde{P} = V \cap \tilde{P} (U, V \in L (C R)),$ for some subclasses...

All completely regular elements in $H y p_{G} (n)$

Ampika Boonmee, Sorasak Leeratanavalee (2013)

Discussiones Mathematicae - General Algebra and Applications

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In Universal Algebra, identities are used to classify algebras into collections, called varieties and hyperidentities are use to classify varieties into collections, called hypervarities. The concept of a hypersubstitution is a tool to study hyperidentities and hypervarieties. Generalized hypersubstitutions and strong identities generalize the concepts of a hypersubstitution and of a hyperidentity, respectively. The set of all generalized hypersubstitutions forms...

Interpolating discrete multiplicity varieties for $A ⁰_{p} (ℂ ⁿ)$

Bao Qin Li, Enrique Villamor (2006)

Studia Mathematica

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A necessary and sufficient condition is obtained for a discrete multiplicity variety to be an interpolating variety for the space $A ⁰_{p} (ℂ ⁿ)$ .

On semigroups of $σ$ -endomorphisms

Iwanik, Anzelm

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The finiteness of compact varieties in $C^{n}$

Walter Rudin (1981)

Annales Polonici Mathematici

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Categorical equivalences of varieties generated by algebras $𝔄$ and $𝔄^{r}$

K. Denecke (1988)

Banach Center Publications

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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...