Semigroups Of Relations Research Articles

On semigroups of relations with the operation of the rectangular product

On semigroups of relations with the operation of the rectangular product

Finitely and non-finitely related words

An algebra is finitely related (or has finite degree) if its term functions are determined by some finite set of finitary relations. Nilpotent monoids built from words, via Rees quotients of free monoids, have been used to exhibit many interesting properties with respect to the finite basis problem. We show that much of this intriguing behaviour extends to the world of finite relatedness by using interlocking patterns called chain, crown, and maelstrom words. In particular, we show that there are large classes of non-finitely related nilpotent monoids that can be used to construct examples of: ascending chains of varieties alternating between finitely and non-finitely related; non-finitely related semigroups whose direct product are finitely related; the addition of an identity element to a non-finitely related semigroup to produce a finitely related semigroup.

Not all nilpotent monoids are finitely related

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

Pointwise stabilization of Bresse systems

Bresse system over the interval (0, L) with pointwise dissipation at ξ∈(0,L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\in (0,{L})$$\\end{document} is analyzed. The exponential stability of the related semigroup is shown provided the dissipative points are of the form ξ∈QL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\in \\mathbb {Q}{L}$$\\end{document} and ξ≠n2m+1L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \ e \\frac{n}{2m+1}L$$\\end{document}, where n,m∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n,m\\in \\mathbb {N}$$\\end{document} and n, and 2m+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2m+1$$\\end{document} are co-prime.

Generating Sets of the Complete Semigroup of Binary Relations Defined by Semilattices of the Class Σ4(X, 4)

Generating Sets of the Complete Semigroup of Binary Relations Defined by Semilattices of the Class Σ4(X, 4)

Generating Sets of the Complete Semigroup of Binary Relations Defined by Semilattices of the Class Σ1(X, 6)

Generating Sets of the Complete Semigroup of Binary Relations Defined by Semilattices of the Class Σ1(X, 6)

Ehresmann–Schein–Nambooripad theorems for classes of biunary semigroups

We obtain an ESN theorem for a very general class of biunary semigroups with idempotent-valued domain and range operations, representing them in terms of small categories equipped with a suitable biaction of the identities on the category. Our results generalise the recent work of Fitzgerald and Kinyon connecting localisable semigroups to transcription categories, as well as that of Lawson linking Ehresmann semigroups to categories with Ehresmann biaction. In contrast to most approaches to ESN theorems, we do not require the categories to be ordered or for their sets of identities to possess any particular structure. Throughout, the biunary semigroups are represented using categories rather than generalised categories of any kind, and we obtain category isomorphisms between the clesses of semigroups and their associated enriched categories, rather than category equivalences. Our results cover the class of DRC-semigroups considered by Jones and Shoufeng Wang, but they also cover cases where not both congruence conditions hold, including examples such as the semigroup of binary relations on a set under demonic composition equipped with domain and range operations.

Formula omitted]-fractional Askey–Wilson integrals and related semigroups of operators

We introduce three one parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey–Wilson operator. We also study these families as families of positive linear approximation operators. Applications include connection relations and bilinear formulas for the Askey–Wilson polynomials, and solving certain dual integral equations.

Representations of Munn algebras and related semigroups

We establish representation types (finite, tame or wild) of finite dimensional Munn algebras with semisimple bases. As an application, we establish representation types of finite Rees matrix semigroups, in particular, 0-simple semigroups, and their mutually annihilating unions.

On Hilfer-Type Fractional Impulsive Differential Equations

Using the Schauder fixed point theorem, we prove the existence of impulsive fractional differential equations using Hilfer fractional derivative and nearly sectorial operators in this paper. We’ve gone over the two scenarios where the related semigroup is compact and noncompact for this purpose. We also go over an example to back up the main points.

Left restriction monoids from left E-completions

Given a monoid S with E any non-empty subset of its idempotents, we present a novel one-sided version of idempotent completion we call left E-completion. In general, the construction yields a one-sided variant of a small category called a constellation by Gould and Hollings. Under certain conditions, this constellation is inductive, meaning that its partial multiplication may be extended to give a left restriction semigroup, a type of unary semigroup whose unary operation models domain. We study the properties of those pairs S,E for which this happens, and characterise those left restriction semigroups that arise as such left E-completions of their submonoid of elements having domain 1. As first applications, we decompose the left restriction semigroup of partial functions on the set X and the right restriction semigroup of left total partitions on X as left and right E-completions respectively of the transformation semigroup TX on X, and decompose the left restriction semigroup of binary relations on X under demonic composition as a left E-completion of the left-total binary relations. In many cases, including these three examples, the construction embeds in a semigroup Zappa-Szép product.

Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. Analyzing singular Dirichlet-to-Neumann operators, we will establish Laurent expansion near singularities as well as Mittag–Leffler expansion for the related quadratic forms. The established results will be exploited to solve definitively the problem of positivity of the related semigroup in the framework of Lebesgue spaces. The obtained results are supported by some examples where we analyze properties of singular Dirichlet-to-Neumann operators related to Neumann and Robin Laplacian on Lipschitz domains. Among other results, we shall demonstrate that regularity of the boundary may affect positivity and derive Mittag-Leffler expansion for the eigenvalues of singular Dirichlet-to-Neumann operators.

A new approach to MGT-thermoviscoelasticity

In this paper we discuss some thermoelastic and thermoviscoelastic models obtained from the Gurtin theory, based on the invariance of the entropy under time reversal. We derive differential systems where the temperature and the velocity are ruled by generalized versions of the Moore-Gibson-Thompson equation. In the one-dimensional case, we provide a complete analysis of the evolution, establishing an existence and uniqueness result valid for any choice of the constitutive parameters. This result turns out to be new also for the MGT equation itself. Then, under suitable assumptions on the parameters, corresponding to the subcritical regime of the system, we prove the exponential stability of the related semigroup.

SEMIGROUPS OF BINARY OPERATIONS AND MAGMA-BASED CRYPTOGRAPHY

In this article, algebras of binary operations as a special case of finitary homogeneous relations algebrasare investigated. The tools of our study are based on unary and associative binary operations acting on theset of ternary relations. These operations are generated by the converse operation and the left-composition ofbinary relations. Using these tools, we are going to define special kinds of ternary relations that correspondto functions, injections, right- and left-total binary relations. Then we obtain criteria for these properties interms of ordered semigroups. Note, that there is an embedding of the semigroup of quasigroups operationsin the semigroup of magmas operation and further in the semigroup of ternary relations. This is similar toembedding the semigroup of bijections in the semigroup of functions and then in the semigroup of binaryrelations. Taking a binary operation as the generator of a cyclic semigroup, we can apply an exponentialsquaring method for the fast computation of its positive integer powers. Given that this is the main methodof public key cryptography, we are adapting the Diffie-Hellman-Merkle key exchange algorithm for magmasas a result.

Left and Right Magnifying Elements in the Semigroup of all Binary Relations

An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = M a]. Let X be a nonempty set and BX the semigroup of binary relations on X. In this paper, we give necessary and sufficient conditions for elements in BX to be left or right magnifying.

Combinatorial characterization of pseudometrics

Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with the domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ is combinatorially similar to $\Psi$ if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y \to X$ such that $\Psi(x, y) = f(\Phi(g(x), g(y)))$ for all $x$, $y \in Y$. It is shown that the semigroups of binary relations generated by sets $\{\Phi^{-1}(a) \colon a \in \Phi(X^{2})\}$ and $\{\Psi^{-1}(b) \colon b \in \Psi(Y^{2})\}$ are isomorphic for combinatorially similar $\Phi$ and $\Psi$. The necessary and sufficient conditions under which a given mapping is combinatorially similar to a pseudometric, or strongly rigid pseudometric, or discrete pseudometric are found. The algebraic structure of semigroups generated by $\{d^{-1}(r) \colon r \in d(X^{2})\}$ is completely described for nondiscrete, strongly rigid pseudometrics and, also, for discrete pseudometrics $d \colon X^{2} \to \mathbb{R}$.

Parametrix method for the first hitting time of an elliptic diffusion with irregular coefficients

ABSTRACT In this article, we are interested in studying the transition density function of the couple given by the first hitting time of a fixed threshold by a one-dimensional uniformly elliptic diffusion process and the associated stopped process. Our strategy relies on the parametrix technique applied to the related semigroup. It notably allows us to extend standard PDEs results on Green and Poisson functions, see e.g. Garroni and Menaldi [Green functions for second order parabolic integro-differential problems, volume 275 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992], by establishing the infinite expansion of the corresponding transition density for irregular coefficients, here bounded measurable drift coefficient and bounded η-Hölder-continuous diffusion coefficient. As a by-product, we establish some Gaussian upper estimates.

О полугруппах отношений с операцией левого и правого прямоугольного произведения

О полугруппах отношений с операцией левого и правого прямоугольного произведения

Ideals of Numerical Semigroups and
 Error-Correcting Codes

Several results relating additive ideals of numerical semigroups and algebraic-geometrycodes are presented. In particular, we deal with the set of non-redundant parity-checks, the codelength, the generalized Hamming weights, and the isometry-dual sequences of algebraic-geometrycodes from the perspective of the related Weierstrass semigroups. These results are related tocryptographic problems such as the wire-tap channel, t-resilient functions, list-decoding, networkcoding, and ramp secret sharing schemes.

SEMIGROUPS GENERATED BY PARTITIONS

Let $X$ be a nonempty set and $X^{2}$ be the Cartesian square of $X$. Some semigroups of binary relations generated partitions of $X^2$ are studied. In particular, the algebraic structure of semigroups generated by the finest partition of $X^{2}$ and, respectively, by the finest symmetric partition of $X^{2}$ are described.