A semilattice of varieties of completely regular semigroups

Mario Petrich

Displaying similar documents to “A semilattice of varieties of completely regular semigroups”

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...

Relations on a lattice of varieties of completely regular semigroups

Mario Petrich (2020)

Mathematica Bohemica

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Completely regular semigroups $𝒞ℛ$ are considered here with the unary operation of inversion within the maximal subgroups of the semigroup. This makes $𝒞ℛ$ a variety; its lattice of subvarieties is denoted by $ℒ (𝒞ℛ)$ . We study here the relations $𝐊, T, L$ and $𝐂$ relative to a sublattice $Ψ$ of $ℒ (𝒞ℛ)$ constructed in a previous publication. For $𝐑$ being any of these relations, we determine the $𝐑$ -classes of all varieties in the lattice $Ψ$ as well as the restrictions of $𝐑$ to $Ψ$ .

On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups

Shrawani Mitkari, Vilas Kharat (2024)

Mathematica Bohemica

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In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices $L (G)$ and sublattices of $L (G)$ . It is proved that the collections of all pronormal subgroups of $A_{n}$ and S $_{n}$ do not form sublattices of respective $L (A_{n})$ and $L (S_{n})$ , whereas the collection of all pronormal subgroups $LPrN ({Dic}_{n})$ of a dicyclic group is a sublattice of $L ({Dic}_{n})$ . Furthermore, it is shown that $L ({Dic}_{n})$ and $LPrN ({Dic}_{n}$ ) are lower semimodular lattices.

Inverses of generators of nonanalytic semigroups

Ralph deLaubenfels (2009)

Studia Mathematica

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Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup ${e^{t A}}_{t \geq 0}$ . It is shown that $A^{- 1}$ generates an $O (1 + τ) A {(1 - A)}^{- 1}$ -regularized semigroup. Several equivalences for $A^{- 1}$ generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of ${e^{t A}}_{t \geq 0}$ , on subspaces, for $A^{- 1}$ generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate...

Regular elements and Green's relations in Menger algebras of terms

Klaus Denecke, Prakit Jampachon (2006)

Discussiones Mathematicae - General Algebra and Applications

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Defining an (n+1)-ary superposition operation $S^{n}$ on the set $W_{τ} (X_{n})$ of all n-ary terms of type τ, one obtains an algebra $n - c l o n e τ : = (W_{τ} (X_{n}); S^{n}, x_{1}, . . ., x_{n})$ of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation $S^{n}$ there are different possibilities to define binary associative operations on the set $W_{τ} (X_{n})$ and on the cartesian power $W_{τ} {(X_{n})}^{n}$ . In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative...

Sufficient conditions for a T-partial order obtained from triangular norms to be a lattice

Lifeng Li, Jianke Zhang, Chang Zhou (2019)

Kybernetika

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For a t-norm T on a bounded lattice $(L, \leq)$ , a partial order $\leq_{T}$ was recently defined and studied. In [11], it was pointed out that the binary relation $\leq_{T}$ is a partial order on $L$ , but $(L, \leq_{T})$ may not be a lattice in general. In this paper, several sufficient conditions under which $(L, \leq_{T})$ is a lattice are given, as an answer to an open problem posed by the authors of [11]. Furthermore, some examples of t-norms on $L$ such that $(L, \leq_{T})$ is a lattice are presented.

Finite Groups with some $s$ -Permutably Embedded and Weakly $s$ -Permutable Subgroups

Fenfang Xie, Jinjin Wang, Jiayi Xia, Guo Zhong (2013)

Confluentes Mathematici

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Let $G$ be a finite group, $p$ the smallest prime dividing the order of $G$ and $P$ a Sylow $p$ -subgroup of $G$ with the smallest generator number $d$ . There is a set $ℳ_{d} (P) = {P_{1}, P_{2}, \dots, P_{d}}$ of maximal subgroups of $P$ such that $⋂_{i = 1}^{d} P_{i} = Φ (P)$ . In the present paper, we investigate the structure of a finite group under the assumption that every member of $ℳ_{d} (P)$ is either $s$ -permutably embedded or weakly $s$ -permutable in $G$ to give criteria for a group to be $p$ -supersolvable or $p$ -nilpotent.

Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices

Gábor Czédli (2024)

Mathematica Bohemica

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Following G. Grätzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no $M_{3}$ as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly irreducible elements and these two elements are complements of each other. A finite poset $P$ is said to be JConSPS-representable if there is an SPS lattice $L$ such that $P$ is isomorphic to the poset $J (Con L)$ of join-irreducible congruences of $L$ . We prove that...

Semigroups generated by convex combinations of several Feller generators in models of mathematical biology

Adam Bobrowski, Radosław Bogucki (2008)

Studia Mathematica

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Let be a locally compact Hausdorff space. Let $A_{i}$ , i = 0,1,...,N, be generators of Feller semigroups in C₀() with related Feller processes $X_{i} = X_{i} (t), t \geq 0$ and let $α_{i}$ , i = 0,...,N, be non-negative continuous functions on with $\sum_{i = 0}^{N} α_{i} = 1$ . Assume that the closure A of $\sum_{k = 0}^{N} α_{k} A_{k}$ defined on $⋂_{i = 0}^{N} (A_{i})$ generates a Feller semigroup T(t), t ≥ 0 in C₀(). A natural interpretation of a related Feller process X = X(t), t ≥ 0 is that it evolves according to the following heuristic rules: conditional on being at a point p ∈ , with probability...

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.

Orthogonality and complementation in the lattice of subspaces of a finite vector space

Ivan Chajda, Helmut Länger (2022)

Mathematica Bohemica

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We investigate the lattice $𝐋 (𝐕)$ of subspaces of an $m$ -dimensional vector space $𝐕$ over a finite field $GF (q)$ with a prime power $q = p^{n}$ together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice $𝐋 (𝐕)$ satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when $𝐋 (𝐕)$ is orthomodular....

A note on sumsets of subgroups in $ℤ *_{p}$

Derrick Hart (2013)

Acta Arithmetica

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Let A be a multiplicative subgroup of $ℤ *_{p}$ . Define the k-fold sumset of A to be $k A = x_{1} + . . . + x_{k} : x_{i} \in A, 1 \leq i \leq k$ . We show that $6 A \supseteq ℤ *_{p}$ for $| A | > p^{11 / 23 + ϵ}$ . In addition, we extend a result of Shkredov to show that ${| 2 A | ≫ | A |}^{8 / 5 - ϵ}$ for $| A | ≪ p^{5 / 9}$ .

$J$ -invariant of linear algebraic groups

Viktor Petrov, Nikita Semenov, Kirill Zainoulline (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $G$ be a semisimple linear algebraic group of inner type over a field $F$ , and let $X$ be a projective homogeneous $G$ -variety such that $G$ splits over the function field of $X$ . We introduce the $J$ -invariant of $G$ which characterizes the motivic behavior of $X$ , and generalizes the $J$ -invariant defined by A. Vishik in the context of quadratic forms. We use this $J$ -invariant to provide motivic decompositions of all generically split projective homogeneous $G$ -varieties, e.g. Severi-Brauer varieties,...