scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

Long unsplittable zero-sum sequences over a finite cyclic group

2016 ◽  
Vol 12 (04) ◽  
pp. 979-993 ◽  
Author(s):  
Pingzhi Yuan ◽  
Yuanlin Li
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Finite Cyclic Group ◽  
Zero Sum

Let [Formula: see text] be an additively written finite cyclic group of order [Formula: see text] and let [Formula: see text] be a minimal zero-sum sequence with elements of [Formula: see text], i.e. the sum of elements of [Formula: see text] is zero, but no proper nontrivial subsequence of [Formula: see text] has sum zero. [Formula: see text] is called unsplittable if there do not exist an element [Formula: see text] in [Formula: see text] and two elements [Formula: see text] in [Formula: see text] such that [Formula: see text] and the new sequence [Formula: see text] is still a minimal zero-sum sequence. In this paper, we investigate long unsplittable minimal zero-sum sequences over [Formula: see text]. Our main result characterizes the structures of all such sequences [Formula: see text] and shows that the index of [Formula: see text] is at most 2, provided that the length of [Formula: see text] is greater than or equal to [Formula: see text] where [Formula: see text] is a positive integer with least prime divisor greater than [Formula: see text].

scholarly journals On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
K. Mageshwaran ◽  
G. Kalaimurugan ◽  
Bussakorn Hammachukiattikul ◽  
Vediyappan Govindan ◽  
Ismail Naci Cangul
Keyword(s):  
Cyclic Group ◽  
Upper Bound ◽  
Labeling Index ◽  
Positive Difference ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
The Difference ◽  
Radio Labeling

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

scholarly journals Automorphisms of Regular Wreath Product -Groups

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Cyclic Group ◽  
Product Group ◽  
Finite Cyclic Group ◽  
Arbitrary Finite Group

We present a useful new characterization of the automorphisms of the regular wreath product group of a finite cyclic -group by a finite cyclic -group, for any prime , and we discuss an application. We also present a short new proof, based on representation theory, for determining the order of the automorphism group Aut(), where is the regular wreath product of a finite cyclic -group by an arbitrary finite -group.

Uniqueness of Free Actions on S3 Respecting a Knot

1987 ◽  
Vol 39 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Michel Boileau ◽  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Group Action ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Cyclic Group Action ◽  
Periodic Diffeomorphisms ◽  
Free Actions

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

scholarly journals Vertex connectivity of the power graph of a finite cyclic group II

2019 ◽  
Vol 19 (02) ◽  
pp. 2050040 ◽  
Author(s):  
Sriparna Chattopadhyay ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Cyclic Group ◽  
Undirected Graph ◽  
Prime Numbers ◽  
Vertex Connectivity ◽  
Induced Subgraph ◽  
Power Graph ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
Positive Integers ◽  
Appl Math

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.

scholarly journals The regular radical of semigroup rings of commutative semigroups

1992 ◽  
Vol 34 (2) ◽  
pp. 133-141 ◽  
Author(s):  
A. V. Kelarev
Keyword(s):  
Regular Semigroup ◽  
Commutative Semigroup ◽  
General Problem ◽  
Associative Ring ◽  
Complete Solution ◽  
Group Rings ◽  
Semigroup Rings ◽  
Regular Group ◽  
Commutative Semigroups

A description of regular group rings is well known (see [12]). Various authors have considered regular semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in semigroup rings for semigroups of the classes mentioned. In [10], the regular semigroup rings of commutative semigroups were described. The aim of the present paper is to characterize the regular radical ρ(R[S]) for each associative ring R and commutative semigroup S.

The Wind-Sand Flux Structure of Grassland on the Northern Foot of Yinshan Mountain

2014 ◽  
Vol 668-669 ◽  
pp. 1530-1537
Author(s):  
Hong Tao Jiang ◽  
Chun Rong Guo ◽  
Chun Xing Hai ◽  
Shan Shan Sun ◽  
Yun Hu Xie ◽  
...  
Keyword(s):  
Vertical Distribution ◽  
Power Function ◽  
Exponential Function ◽  
Vertical Direction ◽  
Power Functions ◽  
Height Range ◽  
Sand Flux ◽  
Soil Flux ◽  
The Vertical Distribution ◽  
Better Than

Sand samplers were layed out in the grassland located in the northern foot of Yinshan Mountain for collecting soil flux samples from 0 to 1.5m height above the surface from Mar., 1, 2008 to Feb., 29, 2009.Exponential and Power functions were both used for describing vertical distribution of sand flux in the grassland, the results indicated that determination coefficient of Power function varied from 0.898 to 0.992 while 0.432 to 0.661 for exponential function. Power function is better than exponential function in describing the vertical distribution of both annual and seasonal soil flux, summer excluded. Annual cumulative percentage of each height was determined indirectly according to the power function mentioned above, the result indicated that up to 2m height,15-25% of soil flux concentrated with in 10cm above the surface,25-35% of soil flux concentrated within 20cm above the surface,30-40% of soil flux concentrated within 30 cm above the surface, 43-54% of soil flux concentrated within 50 cm above the surface,85-90% of soil flux concentrated within 150 cm above the surface, respectively. No significant differences of soil flux structures in spring, autumn, winter and in the whole year were found. The research on wind erosion of grassland in the vertical direction more dispersed, in the height range of sediment accumulated percentage was lower than that of the previous research.

scholarly journals Extended periodic links and HOMFLYPT polynomial

2019 ◽  
Vol 28 (11) ◽  
pp. 1940006
Author(s):  
Nafaa Chbili ◽  
Hajer Jebali
Keyword(s):  
Fixed Points ◽  
Cyclic Group ◽  
Finite Cyclic Group

Extended strongly periodic links have been introduced by Przytycki and Sokolov as a symmetric surgery presentation of three-manifolds on which the finite cyclic group acts without fixed points. The purpose of this paper is to prove that the symmetry of these links is reflected by the first coefficients of the HOMFLYPT polynomial.

scholarly journals A note on local polynomial functions on commutative semigroups

1982 ◽  
Vol 33 (1) ◽  
pp. 50-53
Author(s):  
P. A. Grossman
Keyword(s):  
Positive Integer ◽  
Universal Algebra ◽  
Polynomial Function ◽  
Polynomial Functions ◽  
Commutative Semigroups ◽  
Local Polynomial ◽  
A Chain

AbstractGiven a universal algebra A, one can define for each positive integer n the set of functions on A which can be “interpolated” at any n elements of A by a polynomial function on A. These sets form a chain with respect to inclusion. It is known for several varieties that many of these sets coincide for all algebras A in the variety. We show here that, in contrast with these results, the coincident sets in the chain can to a large extent be specified arbitrarily by suitably choosing A from the variety of commutative semigroups.

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