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TitleOn the Structure of the Subset C(S) of an Ordered Hypersemigroup
Vol.45(6) (2021) page: 789-804
Author(s): F.F. Chen and X.Y. Xie  

In this paper, the notions of the generation and the fundamental of an ordered hypersemigroup are introduced. We obtain the maximal hyperideal generation and the minimal hyperideal generation with relative to the same fundamental of a hyperideal generation. Furthermore, we give the construction of $C(S)$, and find out the least strong congruence on an (ordered) hypersemigroup. As an application of the least strong congruence on an ordered hypersemigroup, we give the description of a contained or anti-contained ordered hypersemigroup. Finally, we give some examples to describe the definitions and properties mentioned in this paper.

Keywords: Ordered hypersemigroup; Generation; Strong congruence; Contained.
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TitleThe Vagner-Preston Representation of a Block-Group
Vol.45(6) (2021) page: 805-812
Author(s): V.H. Fernandes  

In this short note we construct an extension of the Vagner-Preston representation for block-groups and show that its kernel is the largest congruence that separates regular elements.

Keywords: Block-Groups; Representations.
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TitleOn $\sigma$-Subnormal Subgroups of Finite Groups
Vol.45(6) (2021) page: 813-824
Author(s): W. Guo, I.N. Safonova and A.N. Skiba  

Throughout this paper, all groups are finite and $\sigma$ is some partition of the set of all primes $\Bbb{P}$ (that is, $\sigma =\{\sigma_{i} \mid i\in I \}$, where $\Bbb{P}=\cup_{i\in I} \sigma_{i}$ and $\sigma_{i}\cap \sigma_{j}= \emptyset $ for all $i\ne j$). A subgroup $A$ of a group $G$ is said to be \emph{$\sigma$-subnormal} in $G$ if there is a subgroup chain $A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$ such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a $\sigma_{j} $-group for some $j=j(i)$ for all $i=1, \ldots , n$.
In this review, we discuss some known results of the theory of $\sigma$-subnormal subgroups and also some open questions in this line research.

Keywords: Finite group; $\sigma$-soluble group; $\sigma$-nilpotent group; $P\sigma T$-group; $\sigma$-subnormal subgroup.
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TitleOn Automorphisms of a Finite Group $G$ with a Non-trivial Nilpotent Normal Subgroup
Vol.45(6) (2021) page: 825-832
Author(s): L.L. Han and X.Y. Guo  

Let $G$ be a finite group with a nontrivial nilpotent normal subgroup $N$. In this paper, it is shown that under some conditions, every Coleman automorphism or class-preserving Coleman automorphism of $G$ is inner and therefore $G$ has the normalizer property.

Keywords: Class-preserving automorphism; Coleman automorphism; Normalizer property.
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TitleOn k-Comma Codes
Vol.45(6) (2021) page: 833-848
Author(s): H.Y. Liu, K.P. Shum and J. Leng  

Let $L$ be a nonempty language over an alphabet $A$ and $k\geq0$. Then, we call $L$ a $k$-comma code if $L$ satisfies $LA^kL\cap A^+LA^+=\emptyset$. It is obvious that the $k$-comma codes are a generalization of comma-free codes. Our aim in this paper is to further study the properties of the $k$-comma codes. As one can easily observe that the family of infix codes is not closed under composition and therefore, we concentrate on the closure properties of the $k$-comma codes under composition and decomposition. Consequently, we give a sufficient and necessary condition under which the concatenation of two disjoint $k$-comma codes $X$ and $Y$ is still a $k$-comma code under the assumption that $X\cup Y$ is an infix code. Obviously, the $1$-$1$-comma codes are a generalization of $1$-comma codes. Thus, by using the structure properties of $1$-$1$-comma codes, we show that the $1$-$1$-comma property is decidable for regular languages. Finally, we characterize the automata which accept the words in $X_1$ where $X_1 = \set{u \in A^+ | (~\forall w\in A^k)uwu \cap A^+uA^+ = \emptyset}$.

Keywords: Decidability; Automaton; Combinatorics of words; $k$-Comma code.
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TitleMaximal Prefix Codes with Constant Average Length
Vol.45(6) (2021) page: 849-872
Author(s): Y. Liu and R.Y. Li  

In this paper, maximal prefix codes with constant average length are investigated. This class of codes is a generalized class of thin codes. Some basic properties of the class of codes are given in this paper. This class of codes is shown to be closed under two operations ``product of languages'' and ``composition of codes''. Finite maximal prefix codes over binary alphabets with constant average length $n=3$ are completely determined (the cases $n=1$ and $2$ are trivial). There are infinitely many such codes which can be classified into $9$ series.

Keywords: Maximal prefix code; Maximal bifix code; Average length; Bernoulli distribution; Bernoulli set.
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TitleDyad Fuzzy $\beta$-Covering Rough Set Models Based on Fuzzy Information System and Its Generalization over Fuzzy Lattice
Vol.45(6) (2021) page: 873-896
Author(s): X.L. Niu, T.Q. Zhou, Z.D. Sun and X.Z. Kong  

In this paper, we present some aspects in fuzzy decision making theory which is closely related to our daily life applications. In fact, if only one factor is considered when making a decision, it usually leads to one-sided and limited judgment. In order to deal with this problem, we use fuzzy $\beta$-co-neighborhood to define a new fuzzy $\beta$-covering rough set model based on information system for the first time. At the same time, we extend the concept of twin approximation operators to the information system, and propose a new model—dyad fuzzy $\beta$-covering rough set models, which is a combination of covering rough set theory, fuzzy rough set theory and fuzzy information system. This model can analyze and solve practical problems from positive and negative aspects, so as to make more accurate decisions. Therefore, we give an example in medicine to illustrate its practical value. Finally, we generalize the above model to fuzzy lattice and construct $L$-dyad fuzzy $\beta$-covering rough set models.

Keywords: Fuzzy $\beta$-covering; Fuzzy information system; Fuzzy rough set; Fuzzy lattice.
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TitleOn Green's Relations Which Are Related to an Algebraic System of Type ((n);(m))
Vol.45(6) (2021) page: 897-904
Author(s): J. Joomwong, D. Phusanga, S. Jino and J. Koppitz  

Green's relations are five equivalence relations which defined on semigroup or monoid. In the present paper, we consider a semigroup with the Cartesian product of the set of terms and the set of atomic formulas as universe. This semigroup extends the idea of the study of semigroups of terms of a given type. We characterize the Green's relations for that semigroup.

Keywords: Algebraic system; Term; Formula; Relational term; Generalized superposition; Green's relation.
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TitleDiscrete Normal Categories and Cross Connections
Vol.45(6) (2021) page: 905-918
Author(s): A.R. Rajan  

The category of principal left[or right] ideals of a regular semigroup has been abstractly characterised as a normal category by KSS Nambooripad. Cross connection is a relation connecting two normal categories $\categc$ and $\categd$ so that $\categc$ is isomorphic to the normal category of principal left ideals and $\categd$ is isomorphic to the normal category of principal right ideals of a regular semigroup. Here we describe all cross connections of certain categories with a simplified structure. The categories considered are normal categories with the property that the partial order on the set of objects is the identity relation. Such normal categories are said to be discrete normal categories.

Keywords: Normal categories; Normal cone; Cross connection dual; Normal dual; Discrete normal categories; Cross connection semigroup.
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TitleSome Aspects of Semirings
Vol.45(6) (2021) page: 919-930
Author(s): M.K. Sen, S.K. Maity and K.P. Shum  

This article gives a survey of some results on the completely regular semirings, Clifford semirings, Clifford semifields and Leavitt path algebra over Clifford semifield.

Keywords: Completely regular semiring; Clifford semiring; Generalized Clifford semiring; b-lattice; Clifford semifield; Leavitt path algebra.
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TitleGE-Morphisms of GE-Algebras
Vol.45(6) (2021) page: 931-944
Author(s): M.K. Shaik, R. Bandaru and Y.B. Jun  

In this paper, the notion of GE-morphism is introduced, and established fundamental GE-morphism theorem. We investigated some isomorphism theorems in GE-algebras. We derived that the image of a GE-filter of a given GE-algebra $X$ under GE-epimorphism $f:X\rightarrow Y$ is also a GE-filter of a GE-algebra $Y$. We established the one-to-one correspondence between the GE-filters of a GE-algebra $X$ containing $ker(f)$ and GE-filters of $Y$ whenever $f:X\rightarrow Y$ is GE-epimorphism. We introduced the concept of extended GE-morphism(eGE-morphism) in (weak) eGE-algebras and investigated its properties. We provided conditions for the image of superset $F_{X}$ of $E_{X}$ is an eGE-filter of $f(X)$ whenever $f:(X,*_{X},E_{X})\rightarrow (Y,*_{Y},E_{Y})$ is a GE-morphism.

Keywords: GE-algebra; (Weak) eGE-algebra; GE-morphism; eGE-morphism; Kernel.
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TitleOn the Widths of Finite Groups
Vol.45(6) (2021) page: 945-951
Author(s): W.J. Shi  

Let $G$ be a finite group, $\pi(G)$ be the set of prime divisors dividing the order of $G$ and $\pi_{e}(G)$ (spectrum) denote the set of element orders of $G$. Then, we define $w_{o}(G)= |\pi(G)|$ the width of order of $G$ and $w_{s}(G)= max \{|\pi(k)| \mid k \in \pi_{e}(G)\}$ the width of spectrum of $G$. In this paper, we discuss the case of small $w_{o}(G)$, where $G$ is a finite simple group, prove a new results about the two widths of groups. This paper is a part of the revised version based on the author's plenary talk at ``2020 Ural Workshop on Group Theory and Combinatorics".

Keywords: Finite groups; Quantitative characterization; Width of order; Width of spectrum.
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TitleA Brief Survey on Semigroups and Semihypergroups
Vol.45(6) (2021) page: 953-967
Author(s): K.P. Shum and S. Ostadhadi-Dehkordi  

In this survey article, our aim is to give an overview of several substantial developments in the theory of semigroups and semihypergroups. Thus, we shed light on their impact on other branch of mathematics and sciences.

Keywords: Semigroups; Groups; Rings; Semihypergroups; $H_v$-Groups; Ress theorem; Differential rings; Fredholm integral.
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TitleOn Languages Defined by Generalized Principal Right Congruences
Vol.45(6) (2021) page: 969-975
Author(s): S.F. Wang  

The study of languages defined by generalized principal right congruences on a free monoid generated by a finite alphabet was initiated by Prodinger in 1980. In this paper, we introduce some kinds of semifilters related to suffix-closed languages and investigate languages defined by generalized principal right congruences associated with these semifilters. As an application, a characterization of regular languages is reobtained.

Keywords: Semifilters; Suffix-closed languages; Regular languages.
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TitleOn Several Special and General Versions of L.K. Hua's Theorem
Vol.45(6) (2021) page: 977-990
Author(s): S.H. Zheng and K.P. Shum  

In this paper, we give several special and general versions of the well known theorem of L.K. Hua given in 1949. Then, we concentrate on Hua-like problems on semigroups and its related topics. Several simplified proofs of these new versions of Hua's problem in semigroups and their background materials are presented. Recent development on Hua-like problem in semigroups and its related topics are provided and discussed.

Keywords: Hua's Theorem; Hua-like theorem; $LR$-normal orthogroups; $LR$-regular orthogroups.
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TitleEndomorphisms of Certain Semigroups of Finite Full Transformations
Vol.45(6) (2021) page: 991-998
Author(s): X.L. Yang  

Let $T_{n}$ be the semigroup of all full transformations on a finite set $X_{n}=\{1,\dots, n\}$. In this paper, we describe the endomorphisms of the subsemigroups $K(n,r) =\{f\in T_{n}: |im(f)|\leq r\}$ for $1\leq r\leq n-1$.

Keywords: Full transformation semigroups; Proper ideals; Endomorphisms.
Size:107KB   download
TitleContents for Volume 45
Vol.45(6) (2021) page: 999-1003
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