[unknown]

Klaus Denecke; Jörg Koppitz; Nittiya Pabhapote

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A semilattice of varieties of completely regular semigroups

Mario Petrich (2020)

Mathematica Bohemica

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Completely regular semigroups are unions of their (maximal) subgroups with the unary operation within their maximal subgroups. As such they form a variety whose lattice of subvarieties is denoted by $ℒ (𝒞 ℛ)$ . We construct a 60-element $\cap$ -subsemilattice and a 38-element sublattice of $ℒ (𝒞 ℛ)$ . The bulk of the paper consists in establishing the necessary joins for which it uses Polák’s theorem.

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 $A ⁰_{p} (ℂ ⁿ)$ .

Bases for certain varieties of completely regular semigroups

Mario Petrich (2021)

Commentationes Mathematicae Universitatis Carolinae

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Completely regular semigroups equipped with the unary operation of inversion within their maximal subgroups form a variety, denoted by $𝒞ℛ$ . The lattice of subvarieties of $𝒞ℛ$ is denoted by $ℒ (𝒞ℛ)$ . For each variety in an $⋂$ -subsemilattice $Γ$ of $ℒ (𝒞ℛ)$ , we construct at least one basis of identities, and for some important varieties, several. We single out certain remarkable types of bases of general interest. As an application for the local relation $L$ , we construct $𝐋$ -classes of all varieties in $Γ$ . Two...

On a probabilistic problem on finite semigroups

Attila Nagy, Csaba Tóth (2023)

Commentationes Mathematicae Universitatis Carolinae

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We deal with the following problem: how does the structure of a finite semigroup $S$ depend on the probability that two elements selected at random from $S$ , with replacement, define the same inner right translation of $S$ . We solve a subcase of this problem. As the main result of the paper, we show how to construct not necessarily finite medial semigroups in which the index of the kernel of the right regular representation equals two.

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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