Automorphisms of the endomorphism semigroup of a free algebra

2015 ◽  
Vol 25 (08) ◽  
pp. 1223-1238 ◽  
Author(s):  
Xiaosong Sun
Keyword(s):  
Free Algebra ◽  
Poisson Algebra ◽  
New Method ◽  
Arbitrary Field ◽  
Variety Of Algebras ◽  
Endomorphism Semigroup

We suggest a new method for describing automorphisms of the endomorphism semigroup of a free algebra in a variety of algebras. As an application, we describe automorphisms of the endomorphism semigroup of a free Poisson algebra over an arbitrary field [Formula: see text].

Toward Efficient Multiplication Algorithms over Finite Fields in Lagrange Representation

2010 ◽  
Vol 20-23 ◽  
pp. 323-327
Author(s):  
Ming Long Qi ◽  
Luo Zhong ◽  
Qing Ping Guo
Keyword(s):  
Finite Fields ◽  
New Method ◽  
Arbitrary Field ◽  
Modular Multiplication ◽  
Multiplication Algorithm ◽  
Representative Theory

In this paper, we present a representative theory for finite fields called the Lagrange Representation recently initialized by Bajard et al. Our contribution is of introducing a new method for computing the leading coefficient of an arbitrary field polynomial, and establishing a field modular multiplication algorithm. Some concrete examples are given in order to emphasize illustration of the method.

scholarly journals The persistence of universal formulae in free algebras

1987 ◽  
Vol 36 (1) ◽  
pp. 11-17 ◽  
Author(s):  
Anthony M. Gaglione ◽  
Dennis Spellman
Keyword(s):  
Wreath Product ◽  
Free Algebra ◽  
Variety Of Algebras ◽  
Free Algebras ◽  
Product Construction

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements:(1) Fr(V) discriminates V. (1*) The Fs(V) satisfy the same universal sentences for all s≥r. Moreover, we have introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent:(2) Fr(V) strongly discriminates V. (2*) The Fs(V) satisfy the same universal formulas for all s ≥ r whenever elements of Fr(V) are substituted for the unquantified variables. On the surface (2) and (2*) appear to be stronger conditions than (1) and (1*). However, we have shown that for particular varieties (of groups) (2) and (2*) are no stronger than (1) and (1*).

STABLE EQUIVALENCE PROBLEMS FOR FREE ALGEBRAS WITH THE NIELSEN-SCHREIER PROPERTY

2001 ◽  
Vol 11 (06) ◽  
pp. 779-786 ◽  
Author(s):  
ALEXANDER A. MIKHALEV ◽  
JIE-TAI YU
Keyword(s):  
Free Algebra ◽  
Maximal Rank ◽  
Variety Of Algebras ◽  
Free Algebras ◽  
Stable Equivalence ◽  
Two Systems ◽  
Equivalence Problems ◽  
Injective Endomorphism

A variety of algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. For free algebras of finite ranks of Schreier varieties we prove that if two systems of elements are stably equivalent, then they are equivalent. We define the rank of an endomorphism of a free algebra of a Schreier variety and prove that an injective endomorphism of maximal rank does not change the rank of elements of maximal rank.

scholarly journals A New Method for Coupling Random Fields

2002 ◽  
Vol 5 ◽  
pp. 77-94 ◽  
Author(s):  
L. A. Breyer ◽  
G. O. Roberts
Keyword(s):  
Markov Chain ◽  
Random Fields ◽  
New Method ◽  
Stochastic Flow ◽  
Arbitrary Field ◽  
Initial Value ◽  
Sample Paths

AbstractGiven a Markov chain, a stochastic flow that simultaneously constructs sample paths started at each possible initial value can be constructed as a composition of random fields. Here, a method is described for coupling flows by modifying an arbitrary field (consistent with the Markov chain of interest) by an independence Metropolis-Hastings iteration. The resulting stochastic flow is shown to have many desirable coalescence properties, regardless of the form of the original flow.

A Universal Variety of Point Algebras

2010 ◽  
Vol 17 (04) ◽  
pp. 647-658
Author(s):  
R. Padmanabhan ◽  
P. Penner
Keyword(s):  
Free Algebra ◽  
Equational Class ◽  
Variety Of Algebras ◽  
Algebraic Systems ◽  
Universal Variety ◽  
Ordered Pairs

Point algebras introduced by Evans are algebraic systems which capture the essence of multiplications (a,b) · (c,d)=(p,q) defined on the set of all ordered pairs of elements of a set S, where p and q are selected from among a,b,c,d by some well-defined rule. In 1961, Jonsson and Tarski gave an interesting example of a variety of algebras of type 〈2,1,1〉 for illustrating the failure of certain free algebra properties. In this paper, we show that this equational class of algebras, called the JT-variety, is a universal variety of point algebras in the sense that every variety generated by a point algebra is a reduct of the JT-variety.

PRIMITIVE ELEMENTS OF FREE METABELIAN ALGEBRAS OF RANK TWO

2003 ◽  
Vol 13 (01) ◽  
pp. 17-33 ◽  
Author(s):  
VESSELIN DRENSKY ◽  
JIE-TAI YU
Keyword(s):  
Lie Algebras ◽  
Free Algebra ◽  
Primitive Element ◽  
Variety Of Algebras ◽  
Primitive Elements ◽  
Associative Algebras ◽  
Polynomial Algebras ◽  
Rank 2 ◽  
Metabelian Algebras ◽  
Injective Endomorphism

Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.

The automorphisms of endomorphism semigroups of relatively free groups

2018 ◽  
Vol 28 (02) ◽  
pp. 207-215
Author(s):  
V. S. Atabekyan ◽  
H. T. Aslanyan
Keyword(s):  
Free Algebra ◽  
General Result ◽  
Free Groups ◽  
Endomorphism Semigroup ◽  
Burnside Groups ◽  
Inner Automorphisms

The question of describing the automorphisms of [Formula: see text] for a free algebra [Formula: see text] in a certain variety was considered by different authors since 2002. In this paper, we consider this question for the class of all relatively free groups having only cyclic centralizers of non-trivial elements. We prove that each automorphism of the endomorphism semigroup [Formula: see text] of groups [Formula: see text] from this class is uniquely determined by its action on the subgroup of inner automorphisms [Formula: see text]. The obtained general result includes the following cases: absolutely free groups, free Burnside groups of odd period [Formula: see text], free groups of infinitely based varieties of Adian (the cardinality of the set of such varieties is continuum), and so on.

scholarly journals Additive primitive length in relatively free algebras

2019 ◽  
Vol 29 (05) ◽  
pp. 849-859
Author(s):  
Vesselin Drensky
Keyword(s):  
Recent Result ◽  
Lie Algebras ◽  
Free Algebra ◽  
Upper Bound ◽  
Polynomial Algebra ◽  
Variety Of Algebras ◽  
Free Algebras ◽  
Primitive Elements ◽  
Number Formula ◽  
Algebra Over A Field

The additive primitive length of an element [Formula: see text] of a relatively free algebra [Formula: see text] in a variety of algebras [Formula: see text] is equal to the minimal number [Formula: see text] such that [Formula: see text] can be presented as a sum of [Formula: see text] primitive elements. We give an upper bound for the additive primitive length of the elements in the [Formula: see text]-generated polynomial algebra over a field of characteristic 0, [Formula: see text]. The bound depends on [Formula: see text] and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free [Formula: see text]-generated nilpotent-by-abelian Lie algebras is bounded by 5 for [Formula: see text] and by 6 for [Formula: see text]. If the field has two elements only, then our bounds are 6 for [Formula: see text] and 7 for [Formula: see text]. This generalizes a recent result of Ela Aydın for two-generated free metabelian Lie algebras. In all cases considered in the paper, the presentation of the elements as sums of primitive elements can be found effectively in polynomial time.

Selective Sampling and Orientation of Non-Homogeneous Tissues

1968 ◽  
Vol 26 ◽  
pp. 98-99
Author(s):  
C. C. Clawson ◽  
L. W. Anderson ◽  
R. A. Good
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Light Microscopy ◽  
Random Sampling ◽  
New Method ◽  
Microscope Examination ◽  
Small Areas ◽  
Selective Sampling ◽  
Large Numbers ◽  
Specific Orientation

Investigations which require electron microscope examination of a few specific areas of non-homogeneous tissues make random sampling of small blocks an inefficient and unrewarding procedure. Therefore, several investigators have devised methods which allow obtaining sample blocks for electron microscopy from region of tissue previously identified by light microscopy of present here techniques which make possible: 1) sampling tissue for electron microscopy from selected areas previously identified by light microscopy of relatively large pieces of tissue; 2) dehydration and embedding large numbers of individually identified blocks while keeping each one separate; 3) a new method of maintaining specific orientation of blocks during embedding; 4) special light microscopic staining or fluorescent procedures and electron microscopy on immediately adjacent small areas of tissue.

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