Partial Semigroups Research Articles

Liminary C⁎-algebras with boolean spectrum

Liminary C⁎-algebras with boolean spectrum are the main topic of Cignoli's joint paper with G.A. Elliott and the present author (Cignoli et al., 1993 [9]). Solving the analogue of Kaplansky's problem for these algebras, in that paper it is proved that the Murray von Neumann order of projections is sufficient to uniquely recover the C⁎-algebraic structure. In this paper we continue the study of these algebras. Among others, we prove that the Elliott partial semigroup of any such algebra A is canonically extendible to a locally finite MV-algebraE(A), in the sense that every finite subset of E(A) generates a finite subalgebra of E(A). Further, every extremal state s of K0(A) has the property that s(K0(A)) is a cyclic subgroup of R, and K0(A) has general comparability. We characterize central projections in these C*-algebras as fixpoints for the “eccentricity” partial order ⊑ on E(A) introduced in the present author's paper (Mundici, 2023 [25]).

Globalization of partial actions of semigroups

We propose two universal constructions of globalization of a partial action of a semigroup on a set, satisfying certain conditions which arise in Morita theory of semigroups. One of the constructions is based on the tensor product of a partial semigroup act with the semigroup and generalizes the globalization construction of strong partial actions of monoids due to Megrelishvili and Schröder. It produces the initial object in an appropriate category of globalizations of a given partial action. The other construction involves {textrm{Hom}}-sets and is novel even in the monoid setting. It produces the terminal object in an appropriate category of globalizations. While in the group case the results of the two constructions are isomorphic, they can be far different in the monoid case.

Semigroup property of fractional differential operators and its applications

<p style='text-indent:20px;'>We establish partial semigroup property of families of Riemann-Liouville and Caputo fractional differential operators. Using this result we prove theorems on reduction of multi-term fractional differential systems to single-term and multi-order systems. As an application we obtain existence and uniqueness of solution to multi-term Caputo fractional differential systems.</p>

A study on some combinatorial sets in partial semigroups

In this paper, we investigate the image and preimage of the important combinatorial sets such as central sets, [Formula: see text]-sets, and [Formula: see text]-sets which play an important role in the study of combinatorics under certain partial semigroup homomorphism. Using that we prove certain results that deal with the existence of [Formula: see text]-set which are not central in partial semigroup framework.

Skew Pairs of Idempotents in Partial Transformation Semigroups

Let S be a regular semigroup. A pair (e,f) of idempotents of S is said to be a skew pair of idempotents if fe is idempotent, but ef is not. T. S. Blyth and M. H. Almeida (T. S. Blyth and M. H. Almeida, skew pair of idempotents in transformation semigroups, Acta Math. Sin. (English Series), 22 (2006), 1705–1714) gave a characterization of four types of skew pairs—those that are strong, left regular, right regular, and discrete—existing in a full transformation semigroup T(X). In this paper, we do in this line for partial transformation semigroups.

NUMERICAL AND GRAPHICAL RESULTS OF FINITE SYMMETRIC INVERSE (I_{n}) AND FULL (T_{n}) TRANSFORMATION SEMIGROUPS

Supposed is a finite set, then a function is called a finite partial transformation semigroup , which moves elements of from its domain to its co-domain by a distance of where . The total work done by the function is therefore the sum of these distances. It is a known fact that and . In this this research paper, we have mainly presented the numerical solutions of the total work done, the average work done by functions on the finite symmetric inverse semigroup of degree , and the finite full transformation semigroup of degree , as well as their respective powers for a given fixed time in space. We used an effective methodology and valid combinatorial results to generalize the total work done, the average work done and powers of each of the transformation semigroups. The generalized results were tested by substituting small values of and a specified fixed times in space. Graphs were plotted in each case to illustrate the nature of the total work done and the average work done. The results obtained in this research article have an important application in some branch of physics and theoretical computer science

Density in partial semigroups

This short note deals with a result in partial semigroups. We introduce a notion of density in partial semigroups and verify that under a certain condition on a partial semigroup [Formula: see text], this density of a central set in [Formula: see text] is positive and also, the density of a [Formula: see text]-thick subset of [Formula: see text] is always a unit.

Left and right magnifying elements in some generalized partial transformation semigroups

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 = Ma]. Let P(X) be a semigroup of all partial transformations on a set X under the composition of maps. A number of results concerning the necessary and sufficient conditions for elements in some interesting generalized semigroups of partial transformations to be left or right magnifiers are presented.

The semigroup of linear terms

The set of linear terms, i.e. terms in which each variable occurs at most once, does not form a subsemigroup of the so-called diagonal semigroup. We consider the reduct of the diagonal semigroup to the linear terms, which is not a partial semigroup. We extend the set of linear terms by an expression “[Formula: see text]”, that is formally a linear term, obtaining a semigroup. The algebraic structure of this semigroup will be studied in this paper. We characterize the Green’s relations and the regular elements as well as the idempotent elements. Moreover, we discuss the ideal structure.

Relative ranks of some partial transformation semigroups

Relative ranks of some partial transformation semigroups

Actions on semigroups and an infinitary Gowers–Hales–Jewett Ramsey theorem

We introduce the notion of (Ramsey) action on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales–Jewett theorems (for both located and nonlocated words), and the Farah–Hindman–McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken–Taylor theorem of Bergelson–Hindman–Williams as a particular case. We present applications of our Ramsey-theoretic results to the structure of recurrence sets in amenable groups.

Partial semigroups and fuzzy sets

One studies HX-partial semigroups, calculating the fuzzy grade of the associated semi-hypergroups.

A generalized central sets theorem in partial semigroups

The most powerful formulation of the Central Sets Theorem in an arbitrary semigroup was proved in the work of De, Hindman, and Strauss. The sets which satisfy the conclusion of the above Central Sets Theorem are called C-sets. The original Central Sets Theorem was extended by J. McLeod for adequate commutative partial semigroups. In this work, we will extend the Central Sets Theorem obtained by taking all possible adequate sequences in a commutative adequate partial semigroup. We shall also discuss a sufficient condition for being a set C-set in our context.

On Magnifying Elements in E-Preserving Partial Transformation Semigroups

Let S be a semigroup. An element a of S is called a right [left] magnifying element if there exists a proper subset M of S satisfying S = M a [ S = a M ] . Let E be an equivalence relation on a nonempty set X. In this paper, we consider the semigroup P ( X , E ) consisting of all E-preserving partial transformations, which is a subsemigroup of the partial transformation semigroup P ( X ) . The main propose of this paper is to show the necessary and sufficient conditions for elements in P ( X , E ) to be right or left magnifying.

English

English

Left (right) magnifying elements of a partial transformation semigroup

An element [Formula: see text] of a semigroup [Formula: see text] is a called a left (respectively right) magnifying element of [Formula: see text] if [Formula: see text] (respectively [Formula: see text]) for some proper subset [Formula: see text] of [Formula: see text]. In this paper, left magnifying elements and right magnifying elements of a partial transformation semigroup will be characterized. The results obtained generalize the results of Magill [K. D. Magill, Magnifying elements of transformation semigroups, Semigroup Forum, 48 (1994) 119–126].

Ranks and isomorphism theorems of semigroups of linear transformations with restricted range

Let P(V) be the partial linear transformation semigroup of a vector space V under composition. Given a fixed subspace W of V, define the following subsemigroups of P(V): $$\begin{aligned} PT(V,W)&=\{\alpha \in P(V)\ |\ V\alpha \subseteq W\},\\ T(V,W)&=\{\alpha \in P(V,W)\ |\ \mathrm{dom}\ \alpha =V\},\\ I(V,W)&=\{\alpha \in P(V,W)\ |\ \alpha \text { is injective}\}. \end{aligned}$$ In this paper, we prove certain isomorphism theorems and compute the ranks of these three semigroups for any proper subspace W of V when V is a finite-dimensional vector space over a finite field. Gaussian binomial coefficients play an essential role in these computations.

ПРОДОЛЖЕНИЕ ЧАСТИЧНОЙ ПОЛУГРУППОВОЙ ОПЕРАЦИИ

Исследуется внутреннее продолжение частичных операций на частичной полугруппе. Рассматриваются продолжения частичной полугруппы внутренним образом до полной с сохранением ассоциативности. Рассмотрены возможности продолжения частичной операции на частичной полугруппе ненулевых элементов вполне 0-простой полугруппы стандартным и нестандартным способами. Получен отрицательный ответ на вопрос о том, любое ли продолжение частичной операции на частичной полугруппе ненулевых элементов вполне 0-простой полугруппы является вполне простой полугруппой, и всякое ли продолжение стандартно. Однако в некоторых случаях ответы положительные. Получены необходимые и достаточные условия продолжаемости частичной операции на частичной полугруппе ненулевых вычетов, а также частичной операции на полугруппе ненулевых (2 × 2)-матриц над полем. Показана единственность продолжения частичной операции на полугруппе ненулевых (2 × 2)-матриц над полем. We analyse inner extensions of partial operations on a partial semigroup. The problem of extension of a partial operation internally to a full one with preservation of associativity is studied. The possibilities of continuing a partial operation on a partial semigroup of non-zero elements of a completely 0-simple semigroup by standard and non-standard methods are considered. A negative answer is obtained in relation to the question about whether any extension of a partial operation on a partial semigroup of non-zero elements is a completely simple semigroup, and whether any extension is standard. However, in certain cases the answers are positive. The article deduces the necessary and sufficient conditions of extendibility of a partial operation on a semigroup of residue modulo n, and also of a partial operation on a semigroup of non-zero elements of (2 × 2)-matrices over the field. The uniqueness of the extension of a partial operation on the semigroup of non-zero (2 × 2)-matrices over a field is shown.

Post-Quantum Cryptography: generalized ElGamal cipher over GF(251^8)

Post-Quantum Cryptography (PQC) attempts to find cryptographic protocols resistant to attacks by means of for instance Shor’s polynomial time algorithm for numerical field problems like integer factorization (IFP) or the discrete logarithm (DLP). Other aspects are the backdoors discovered in deterministic random generators or recent advances in solving some instances of DLP. The use of alternative algebraic structures like non-commutative or non-associative partial groupoids, magmas, monoids, semigroups, quasigroups or groups, are valid choices for these new kinds of protocols. In this paper, we focus in an asymmetric cipher based on a generalized ElGamal non-arbitrated protocol using a non-commutative general linear group. The developed protocol forces a hard subgroup membership search problem into a non-commutative structure. The protocol involves at first a generalized Diffie-Hellman key interchange and further on the private and public parameters are recursively updated each time a new cipher session is launched. Security is based on a hard variation of the Generalized Symmetric Decomposition Problem (GSDP). Working with $GF(251^8)$ a 64-bits security is achieved, and if $GF(251^{16})$ is chosen, the security rises to 127-bits. An appealing feature is that there is no need for big number libraries as all arithmetic if performed in $\Z_{251}$ and therefore the new protocol is particularly useful for computational platforms with very limited capabilities like smartphones or smartcards.

On algebras of P-Ehresmann semigroups and their associate partial semigroups

P-Ehresmann semigroups are introduced by Jones as a common generalization of Ehresmann semigroups and regular $$*$$ -semigroups. Ehresmann semigroups and their semigroup algebras are investigated by many authors in literature. In particular, Stein shows that under some finiteness condition, the semigroup algebra of an Ehresmann semigroup with a left (or right) restriction condition is isomorphic to the category algebra of the corresponding Ehresmann category. In this paper, we generalize this result to P-Ehresmann semigroups. More precisely, we show that for a left (or right) P-restriction locally Ehresmann P-Ehresmann semigroup $$\mathbf{S}$$ , if its projection set is principally finite, then we can give an algebra isomorphism between the semigroup algebra of $$\mathbf{S}$$ and the partial semigroup algebra of the associate partial semigroup of $$\mathbf{S}$$ . Some interpretations and necessary examples are also provided to show why the above isomorphism dose not work for more general P-Ehresmann semigroups.