Special Semigroups Research Articles

Decomposition of pseudo-uninorms with continuous underlying functions via ordinal sum

The decomposition of all pseudo-uninorms with continuous underlying functions, defined on the unit interval, via Clifford's ordinal sum is described. It is shown that each such pseudo-uninorm can be decomposed into representable and trivial semigroups, and special semigroups defined on two points, where the corresponding semigroup operation is the projection to one of the coordinates. Linear orders, for which the ordinal sum of such semigroups yields a pseudo-uninorm, are also characterized.

On the Pecking Order Between Those of Mitsch and Clifford

ABSTRACT Order-theoretic explorations of algebraic structures are known to lead to hitherto hidden insights. Two such relations that have stood out are those of Mitsch and Clifford – the former for the generality in its application and the latter for the insights it offers. In this work, our motivation is to study the converse: we want to explore the extent of the utility of Mitsch’s order and the applicability of Clifford’s order. Firstly, we show that if the Mitsch’s poset is either bounded or a chain, arguably a richer order theoretic structure, the semigroup reduces to one of a simple band. Secondly, noting that the special semigroups on which Clifford’s relation does give rise to an order has not been characterised so far, we solve this problem by proposing a property called Quasi-Projectivity that is essential in this context and also give necessary and sufficient conditions for the Clifford’s relation to give a total and compatible order, even if the semigroup is not commutative. Further, by showing some interesting connections between this relation and the orders obtained by Green’s relations, we further reaffirm the importance and naturalness of the order proposed by Clifford. Finally, by discussing the Clifford’s relations on ordered semigroups, we present some novel perspectives and also show that some of the assumptions in the often cited results of Clifford’s are not necessary. On the whole, our study argues favourably towards Clifford’s than that of the Mitsch’s relation, in so far as the structural information gained about the underlying semigroup.

Semigroup structures via IVI-octahedron sets

In this paper, we use IVI-octahedron ideals and IVI-octahedron bi-ideals to discuss some characterizations of special semigroups, e.g., an intra-regular semigroup, a semigroup which is semilattice of left [respectively, right] simple semigroups or simple left groups and a semigroup which is a semilattice of groups and a [respectively, left and right] simple semigroups. Finally, we define an IVIO-simple [IVIOL-simple and IVI) R-simple] semigroup and study some of its properties and we give a characterization of a semisimple semigroup by IVIOIs.

Exponential decay of matrix Φ-entropies on Markov semigroups with applications to dynamical evolutions of quantum ensembles

In this work, we extend the theory of quantum Markov processes on a single quantum state to a broader theory that covers Markovian evolution of an ensemble of quantum states, which generalizes Lindblad’s formulation of quantum dynamical semigroups. Our results establish the equivalence between an exponential decrease of the matrix Φ-entropies and the Φ-Sobolev inequalities, which allows us to characterize the dynamical evolution of a quantum ensemble to its equilibrium. In particular, we study the convergence rates of two special semigroups, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix Φ-entropy of the resulting quantum ensemble decays exponentially as time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity—the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix Φ-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. There is no classical analogy for these different equilibrium situations. Finally, we also study a statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes and provide upper bounds to the mixing time.

Frames in Hermite-Bergman and special Hermite-Bergman spaces

It is well known that the image of \(L^2(\mathbb {R}^n)\) under the Hermite semigroup \(e^{-tH}\) is a Bergman space \(\mathcal {H}_t(\mathbb {C}^n)\) with the reproducing kernel \(L_t\), for \(t>0\). In this paper, it is shown that the discrete system \(\{e_{ap+ibq, t}:p,q\in \mathbb {Z}^n\}\), for \(a,b>0\), arising out of \(L_t\) turns out to be a frame for \(\mathcal {H}_t(\mathbb {C}^n)\) iff \(ab<\pi \sinh 4t\). A similar type of problem is also discussed for Bessel sequences and frames in holomorphic Sobolev spaces associated with the Hermite semigroup. An attempt is also made to study a similar type of problem for the Bergman space \(\mathcal {B}_t^*(\mathbb {C}^{2n})\), which is the image of \(L^2(\mathbb {C}^n)\) under the special Hermite semigroup.

Special Semigroup Classes Over Some Monoid Constructions and a New Example of a Finitely Presented Monoid with a Non-Finitely Generated Group of Units

Abstract . In this paper, necessary and sufficient conditions are studied for Bruck-Reilly and gener- alized Bruck-Reilly ∗-extensions of direct product of k monoids to be regular, unit regular, completely regular and orthodox. Moreover, we give an example of a finitely presented monoid (generalized Bruck-Reilly ∗-extension of Bruck-Reilly extension of a free group with infinite rank), the group of units of which is not finitely generated. 2010 Mathematics Subject Classification: 16S15; 20E06; 20E22. Keywords and Phrases: Generalized Bruck-Reilly ∗-extension, finite generation, finite presentability. Ozet. Bu makalede k tane monoidin direkt carpiminin Bruck-Reilly ve genellestirilmis Bruck-Reilly *- genislemelerinin, reguler, terslenebilir reguler, tamamen reguler ve orthodox olabilmesi icin gerek ve yeter kosullar calisilmistir. Ayrica, biz terslenebilir elemanlarinin grubu sonlu uretecli olmayan sonlu sunumlu bir monoid (sonsuz rankli bir serbest grubun Bruck-Reilly genislemesinin genellestirilmis Bruck-Reilly *-genislemesi) ornegi verdik. Anahtar Kelimeler: Genellestirilmis Bruck-Reilly *-genislemesi, sonlu ureteclilik, sonlu sunumluluk.

Weighted Tensor Products of Joyal Species, Graphs, and Charades

Motivated by the weighted Hurwitz product on sequences in an algebra, we produce a family of monoidal structures on the category of Joyal species. We suggest a family of tensor products for charades. We begin by seeing weighted derivational algebras and weighted Rota-Baxter algebras as special monoids and special semigroups, respectively, for the same monoidal structure on the category of graphs in a monoidal additive category. Weighted derivations are lifted to the categorical level.

On the Decay of Bounded Semigroup on the Domain of its Generator

For a bounded C0-semigroup on a Banach space X, we prove the following statement: the rate of decay of the semigroup on the domain of its generator is bounded by some decreasing function if and only if the spectrum of the semigroup does not contain any pure imaginary points. Our approach is based on the analysis of a special semigroup on the space of bounded linear operators ℒ(X, X).

On almost surely bounded semigroups of random linear operators

Menger proposed transferring the probabilistic notions of quantum mechanics to the underlying geometry. Following Menger's idea, the notion of random metric spaces is a random generalization of that of metric spaces and also plays an important role in the study of random operator equations. The main difficulty of this article is to work out a skill to give a property peculiar to a special semigroup of random operators, which is not involved in the classical case. Subsequently, some random operators equations are studied, in particular, we discuss the Schrödinger-type random equation, which considerably generalizes the corresponding result in the sense of Skorohod.

SPECIAL SEMIGROUP AMALGAMS OF QUASI UNITARY SUBSEMIGROUPS AND OF QUASI NORMAL BANDS

We show that the special semigroup amalgam with core as a quasi-unitary subsemigroup is embedded in a semigroup. We also show that the special semigroup amalgam within the class of left [right] quasinormal bands is embeddable in a left [right] quasinormal band.

Generating higher-order Lie algebras by expanding Maurer–Cartan forms

By means of a generalization of the Maurer–Cartan expansion method, we construct a procedure to obtain expanded higher-order Lie algebras. The expanded higher-order Maurer–Cartan equations for the case G=V0⊕V1 are found. A dual formulation for the S-expansion multialgebra procedure is also considered. The expanded higher-order Maurer–Cartan equations are recovered from S-expansion formalism by choosing a special semigroup. This dual method could be useful in finding a generalization to the case of a generalized free differential algebra, which may be relevant for physical applications in, e.g., higher-spin gauge theories.

A special matrix transformation semigroup of primitive pairs and the genealogy of Pythagorean triples

A special matrix semigroup acting on the set of primitive pairs of positive integers naturally parameterizing certainDiophantine equations (such as those defining Pythagorean triples) is constructed and studied. The Pythagorean triples are related to a trichotomy tree of solutions.

On Superabundant Semigroups Whose Set of Idempotents Forms a Subsemigroup

In this paper, we study the structure of a superabundant semigroup S whose set of idempotents E(S) forms a subsemigroup. We call such a semigroup a cyber-group because it is a generalization of orthogroups in the class of completely regular semigroups studied by Petrich and Reilly. We show that a cyber-group can be expressed as a semilattice of rectangular monoids. Thus, our result generalizes the well-known result obtained by Petrich in 1987 for orthogroups. Some properties of cyber-groups are given and some special superabundant semigroups are discussed.

Quasi-C-Ehresmann Semigroups and Their Subclasses

We introduce the concept of a quasi-C-Ehresmann semigroup. Some special quasi-C-Ehresmann semigroups will be considered. We will show that the quasi-C-Ehresmann semigroups are special U-semiabundant semigroups and orthodox U-liberal semigroups, which have been recently studied by several authors. Some characterization theorems of this new class of semigroups are obtained. Many results of quasi-C-semigroups in the literature can be extended to quasi-C-Ehresmann Semigroups.

On E-solid Semigroups with Inverse Transversals

We characterize E-solid semigroups with inverse transversals and some special such semigroups in terms of congruences. Some properties of inverse congruences over completely simple semigroups are obtained.

Asymptotic expansions of the representation formulae for (C0) m-parameter operator semigroups1.2

In this paper, we expand asymptotically the general representation formulae for (C 0 ) m-parameter operator semigroups. When we consider special semigroups, our results yield the asymptotic expansions for multivariate Feller operators. In particular, the asymptotic expansions for univariate and multivariate Bernstein operators are reobtained. See the related examples at the end.

Description of the space of pseudocharacters invariant with respect to a special semigroup of endomorphisms for a free semigroup of rank two

Description of the space of pseudocharacters invariant with respect to a special semigroup of endomorphisms for a free semigroup of rank two

ON THE WORD PROBLEM FOR SPECIAL SEMIGROUPS

An algorithm is constructed which solves the word problem for special semigroups with overlapping of defining words less than 1/3.Bibliography: 4 titles. Figures: 10.

On Decompositions of Radially Symmetric Distributions

Previous article Next article On Decompositions of Radially Symmetric DistributionsL. S. KudinaL. S. Kudinahttps://doi.org/10.1137/1120072PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Yu. V. Linnik and , I. V. Ostrovskii, Decomposition of random variables and random vectors, Nauka, Moscow, 1971 Google Scholar[2] N. I. Akhiezer, The classical moment problem and some related questions in analysis, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965x+253 MR0184042 0135.33803 Google Scholar[3] B. Ya. Levin, The distribution of the zeros of entire functions of several variables, Nauka, Moscow, 1971 Google Scholar[4] L. I. Ronkin, Introduction to the theory of entire functions of several variables, American Mathematical Society, Providence, R.I., 1974vi+273 MR0346175 0286.32004 CrossrefGoogle Scholar[5] I˘. V. Ostrovskii˘, A description of the class $I\sb{0}$ in a special semigroup of probability measures, Dokl. Akad. Nauk SSSR, 209 (1973), 788–791 MR0321147 Google Scholar[6] K. R. Parthasarathy, , R. Ranga Rao and , S. R. S. Varadhan, On the category of indecomposable distributions on topological groups, Trans. Amer. Math. Soc., 102 (1962), 200–217 MR0153041 0104.36205 CrossrefGoogle Scholar[7] L. S. Kudina, Indecomposability of the arcsine law, Dokl. Akad. Nauk SSSR, 204 (1972), 25–26 MR0303575 0258.60020 Google Scholar[8] N. Ya. Sonin, Investigations of Cylinder Functions and Special Polynomials, Gostekizdat, Moscow, 1954 Google Scholar[9] I˘. V. Ostrovskii˘, Description of the class I0 in a certain special semigroup of probability measures, Mathematical physics and functional analysis, No. 4 (Russian), Fiz.-Tehn. Inst. Nizkih Temperatur, Akad. Nauk Ukrain. SSR, Kharkov, 1973, 3–12, 160 MR0494404 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Summary of Papers Presented at Sessions of the Probability and Statistics Section of the Moscow Mathematical Society (November 1975–April 1976)17 July 2006 | Theory of Probability & Its Applications, Vol. 21, No. 4AbstractPDF (438 KB) Volume 20, Issue 3| 1976Theory of Probability & Its Applications History Submitted:02 June 1975Published online:17 July 2006 InformationCopyright © 1976 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1120072Article page range:pp. 644-648ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Special semigroups on the two-cell

Special semigroups on the two-cell