scholarly journals On semigroups and groups of local polynomial functions

1979 ◽  
Vol 28 (2) ◽  
pp. 214-218
Author(s):  
Wilfried Nöbauer
Keyword(s):  
Positive Integer ◽  
Wreath Product ◽  
Direct Product ◽  
Polynomial Function ◽  
Subject Classification ◽  
Polynomial Functions ◽  
Semi Group ◽  
Factor Ring ◽  
Local Polynomial

AbstractLet Zn be the factor ring of the integers mod n and t be a positive integer. In this paper some results are given on the structure of the semigroup of all mappings from Zn into Zn and on the structure of the group of all permutations on Zn, which, for any t elements, can be represented by a polynomial function. If n = ab and a, b are relatively prime, then this (semi)group is isomorphic to the direct product of the respective (semi)groups for a and b. Thus it is sufficient to consider only the case where n = pe, p being a prime. In this case it is proved, that the (semi)group is isomorphic to the wreath product of a certain sub(semi)group of the symmetric (semi)group on Zpe−1 by the symmetric (semi)group on Zp. Some remarks on these sub(semi)groups are given.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 B 99, 13 B 25.

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.

scholarly journals Local polynomial functions on factor rings of the integers

1979 ◽  
Vol 27 (2) ◽  
pp. 232-238 ◽  
Author(s):  
Hans Lausch ◽  
Wilfried Nöbauer
Keyword(s):  
Abelian Group ◽  
Universal Algebra ◽  
Polynomial Function ◽  
Commutative Rings ◽  
Infinite Length ◽  
Polynomial Functions ◽  
Local Polynomial ◽  
Definition Of ◽  
Image Position

AbstractLet A be a universal algebra. A function ϕ Ak-A is called a t-local polynomial function, if ϕ can ve interpolated on any t places of Ak by a polynomial function— for the definition of a polynomial function on A, see Lausch and Nöbauer (1973), Let Pk(A) be the set of the polynomial functions, LkPk(A) the set of all t-local polynmial functions on A and LPk(A) the intersection of all LtPk(A), then . If A is an abelian group, then this chain has at most five distinct members— see Hule and Nöbauer (1977)— and if A is a lattice, then it has at most three distinct members— see Dorninger and Nöbauer (1978). In this paper we show that in the case of commutative rings with identity there does not exist such a bound on the length of the chain and that, in this case, there exist chains of even infinite length.

scholarly journals ON NEAR-RING IDEMPOTENTS AND POLYNOMIALS ON DIRECT PRODUCTS OF Ω-GROUPS

2001 ◽  
Vol 44 (2) ◽  
pp. 379-388 ◽  
Author(s):  
Erhard Aichinger
Keyword(s):  
Direct Product ◽  
Subject Classification ◽  
Idempotent Element ◽  
Direct Products ◽  
Primary 16Y30 ◽  
Secondary 08A40

AbstractLet $N$ be a zero-symmetric near-ring with identity, and let $\sGa$ be a faithful tame $N$-group. We characterize those ideals of $\sGa$ that are the range of some idempotent element of $N$. Using these idempotents, we show that the polynomials on the direct product of the finite $\sOm$-groups $V_1,V_2,\dots,V_n$ can be studied componentwise if and only if $\prod_{i=1}^nV_i$ has no skew congruences.AMS 2000 Mathematics subject classification: Primary 16Y30. Secondary 08A40

Matrix representation of formal polynomials over max-plus algebra

2020 ◽  
pp. 2150216
Author(s):  
Cailu Wang ◽  
Yuegang Tao
Keyword(s):  
Matrix Method ◽  
Polynomial Algorithm ◽  
Polynomial Function ◽  
Matrix Representation ◽  
Sufficient Condition ◽  
Polynomial Functions ◽  
Canonical Forms ◽  
Necessary And Sufficient Condition ◽  
The Matrix ◽  
Necessary And Sufficient

This paper proposes the matrix representation of formal polynomials over max-plus algebra and obtains the maximum and minimum canonical forms of a polynomial function by standardizing this representation into a canonical form. A necessary and sufficient condition for two formal polynomials corresponding to the same polynomial function is derived. Such a matrix method is constructive and intuitive, and leads to a polynomial algorithm for factorization of polynomial functions. Some illustrative examples are presented to demonstrate the results.

scholarly journals Direct product and wreath product of transformation semigroups

2012 ◽  
Vol 31 ◽  
pp. 1-7
Author(s):  
Subrata Majumdar ◽  
Kalyan Kumar Dey ◽  
Mohd Altab Hossain
Keyword(s):  
Wreath Product ◽  
Direct Product ◽  
Transformation Semigroups

In this paper direct product and wreath product of transformation semigroups have been defined, and associativity of both the products and distributivity of wreath product over direct product have been established.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10303GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 1-7

scholarly journals On rings all of whose factor rings are integral domains

1993 ◽  
Vol 55 (3) ◽  
pp. 325-333
Author(s):  
Yasuyuki Hirano
Keyword(s):  
Positive Integer ◽  
Zero Divisor ◽  
Arbitrary Positive Integer ◽  
Integral Domains ◽  
Factor Ring ◽  
Zero Divisors ◽  
Proper Quotient

AbstractA ring R is called a (proper) quotient no-zero-divisor ring if every (proper) nonzero factor ring of R has no zero-divisors. A characterization of a quotient no-zero-divisor ring is given. Using it, the additive groups of quotient no-zero-divisor rings are determined. In addition, for an arbitrary positive integer n, a quotient no-zero-divisor ring with exactly n proper ideals is constructed. Finally, proper quotient no-zero-divisor rings and their additive groups are classified.

scholarly journals The Extrapolar SWIFT model (version 1.0): fast stratospheric ozone chemistry for global climate models

2018 ◽  
Vol 11 (2) ◽  
pp. 753-769 ◽  
Author(s):  
Daniel Kreyling ◽  
Ingo Wohltmann ◽  
Ralph Lehmann ◽  
Markus Rex
Keyword(s):  
Polynomial Function ◽  
Stratospheric Ozone ◽  
Ozone Layer ◽  
Rate Of Change ◽  
Polynomial Functions ◽  
Mixing Ratio ◽  
Fourth Degree ◽  
Stratospheric Chemistry ◽  
Wide Range ◽  
Atlas Model

Abstract. The Extrapolar SWIFT model is a fast ozone chemistry scheme for interactive calculation of the extrapolar stratospheric ozone layer in coupled general circulation models (GCMs). In contrast to the widely used prescribed ozone, the SWIFT ozone layer interacts with the model dynamics and can respond to atmospheric variability or climatological trends. The Extrapolar SWIFT model employs a repro-modelling approach, in which algebraic functions are used to approximate the numerical output of a full stratospheric chemistry and transport model (ATLAS). The full model solves a coupled chemical differential equation system with 55 initial and boundary conditions (mixing ratio of various chemical species and atmospheric parameters). Hence the rate of change of ozone over 24 h is a function of 55 variables. Using covariances between these variables, we can find linear combinations in order to reduce the parameter space to the following nine basic variables: latitude, pressure altitude, temperature, overhead ozone column and the mixing ratio of ozone and of the ozone-depleting families (Cly, Bry, NOy and HOy). We will show that these nine variables are sufficient to characterize the rate of change of ozone. An automated procedure fits a polynomial function of fourth degree to the rate of change of ozone obtained from several simulations with the ATLAS model. One polynomial function is determined per month, which yields the rate of change of ozone over 24 h. A key aspect for the robustness of the Extrapolar SWIFT model is to include a wide range of stratospheric variability in the numerical output of the ATLAS model, also covering atmospheric states that will occur in a future climate (e.g. temperature and meridional circulation changes or reduction of stratospheric chlorine loading). For validation purposes, the Extrapolar SWIFT model has been integrated into the ATLAS model, replacing the full stratospheric chemistry scheme. Simulations with SWIFT in ATLAS have proven that the systematic error is small and does not accumulate during the course of a simulation. In the context of a 10-year simulation, the ozone layer simulated by SWIFT shows a stable annual cycle, with inter-annual variations comparable to the ATLAS model. The application of Extrapolar SWIFT requires the evaluation of polynomial functions with 30–100 terms. Computers can currently calculate such polynomial functions at thousands of model grid points in seconds. SWIFT provides the desired numerical efficiency and computes the ozone layer 104 times faster than the chemistry scheme in the ATLAS CTM.

scholarly journals COMBINATORIAL TECHNIQUES FOR DETERMINING RANK AND IDEMPOTENT RANK OF CERTAIN FINITE SEMIGROUPS

2002 ◽  
Vol 45 (3) ◽  
pp. 617-630 ◽  
Author(s):  
Inessa Levi ◽  
Steve Seif
Keyword(s):  
Positive Integer ◽  
Directed Graph ◽  
Subject Classification ◽  
Generating Set ◽  
Finite Semigroups ◽  
Idempotent Rank ◽  
Strongly Connected ◽  
Primary 20M20

AbstractLet $\tau$ be a partition of the positive integer $n$. A partition of the set $\{1,2,\dots,n\}$ is said to be of type $\tau$ if the sizes of its classes form the partition $\tau$ of $n$. It is known that the semigroup $S(\tau)$, generated by all the transformations with kernels of type $\tau$, is idempotent generated. When $\tau$ has a unique non-singleton class of size $d$, the difficult Middle Levels Conjecture of combinatorics obstructs the application of known techniques for determining the rank and idempotent rank of $S(\tau)$. We further develop existing techniques, associating with a subset $U$ of the set of all idempotents of $S(\tau)$ with kernels of type $\tau$ a directed graph $D(U)$; the directed graph $D(U)$ is strongly connected if and only if $U$ is a generating set for $S(\tau)$, a result which leads to a proof if the fact that the rank and the idempotent rank of $S(\tau)$ are both equal to$$ \max\biggl\{\binom{n}{d},\binom{n}{d+1}\biggr\}. $$AMS 2000 Mathematics subject classification: Primary 20M20; 05A18; 05A17; 05C20

scholarly journals Local polynomial functions on semilattices

1981 ◽  
Vol 69 (2) ◽  
pp. 281-286 ◽  
Author(s):  
P.A Grossman
Keyword(s):  
Polynomial Functions ◽  
Local Polynomial

scholarly journals Properties of Armendariz rings and weak Armendariz rings

2009 ◽  
Vol 85 (99) ◽  
pp. 131-137 ◽  
Author(s):  
Dusan Jokanovic
Keyword(s):  
Power Series ◽  
Direct Product ◽  
Nilpotent Ideal ◽  
Factor Ring ◽  
Ring Isomorphism ◽  
Armendariz Rings ◽  
Armendariz Ring

We consider some properties of Armendariz and rigid rings. We prove that the direct product of rigid (weak rigid), weak Armendariz rings is a rigid (weak rigid), weak Armendariz ring. On the assumption that the factor ring R/I is weak Armendariz, where I is nilpotent ideal, we prove that R is a weak Armendariz ring. We also prove that every ring isomorphism preserves weak skew Armendariz structure. Armendariz rings of Laurent power series are also considered.

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