scholarly journals ℤ2-graded codimensions of unital algebras

2018 ◽  
Vol 28 (03) ◽  
pp. 483-500
Author(s):  
Dušan D. Repovš ◽  
Mikhail V. Zaicev
Keyword(s):  
Polynomial Identities ◽  
Combinatorial Properties ◽  
Nonassociative Algebras ◽  
Graded Identities ◽  
Sturmian Words

We study polynomial identities of nonassociative algebras constructed by using infinite binary words and their combinatorial properties. Infinite periodic and Sturmian words were first applied for constructing examples of algebras with an arbitrary real PI-exponent greater than one. Later, we used these algebras for a confirmation of the conjecture that PI-exponent increases precisely by one after adjoining an external unit to a given algebra. Here, we prove the same result for these algebras for graded identities and graded PI-exponent, provided that the grading group is cyclic of order two.

Identities on algebras and combinatorial properties of binary words

2019 ◽  
Vol 489 (5) ◽  
pp. 449-451
Author(s):  
M. V. Zacicev ◽  
D. D. Repovs
Keyword(s):  
Asymptotic Behavior ◽  
Combinatorial Complexity ◽  
Finitely Generated ◽  
Polynomial Identities ◽  
New Approach ◽  
Combinatorial Properties ◽  
Binary Word ◽  
Nonassociative Algebras

We consider polynomial identities and codimension growth of nonassociative algebras over a field of characte-ristics zero. We offer new approach which allows to construct nonassociative algebras starting from a given infinite binary word. The sequence of codimensions of such an algebra is closeely connected with combinatorial complexity of the defining word. These constructions give new examples of algebras with abnormal codimension growth. The first important achievement is that our algebras are finitely generated. The second one is that asymptotic behavior of codimension sequences is quite different unlike all previous examples.

scholarly journals RECENT RESULTS ON EXTENSIONS OF STURMIAN WORDS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 371-385 ◽  
Author(s):  
JEAN BERSTEL
Keyword(s):  
Infinite Words ◽  
Combinatorial Properties ◽  
The Past ◽  
Letter Alphabet ◽  
Sturmian Words

Sturmian words are infinite words over a two-letter alphabet that admit a great number of equivalent definitions. Most of them have been given in the past ten years. Among several extensions of Sturmian words to larger alphabets, the Arnoux–Rauzy words appear to share many of the properties of Sturmian words. In this survey, combinatorial properties of these two families are considered and compared.

COMBINATORIAL PROPERTIES OF STURMIAN PALINDROMES

2006 ◽  
Vol 17 (03) ◽  
pp. 557-573 ◽  
Author(s):  
ALDO DE LUCA ◽  
ALESSANDRO DE LUCA
Keyword(s):  
Combinatorial Properties ◽  
Sturmian Words

We study some structural and combinatorial properties of Sturmian palindromes, i.e., palindromic finite factors of Sturmian words. In particular, we give a formula which permits to compute in an exact way the number of Sturmian palindromes of any length. Moreover, an interesting characterization of Sturmian palindromes is obtained.

GRADED POLYNOMIAL IDENTITIES FOR THE LIE ALGEBRA sl2(K)

2008 ◽  
Vol 18 (05) ◽  
pp. 825-836 ◽  
Author(s):  
PLAMEN KOSHLUKOV
Keyword(s):  
Lie Algebra ◽  
Cyclic Group ◽  
Invariant Theory ◽  
Base Field ◽  
Polynomial Identities ◽  
Graded Identities ◽  
Graded Polynomial Identities

The Lie algebra sl2(K) over a field K has a natural grading by ℤ2, the cyclic group of order 2. We describe the graded polynomial identities for this grading when the base field is infinite and of characteristic different from 2. We exhibit a basis of these identities that consists of one polynomial. In order to obtain this basis we employ methods and results from Invariant theory. As a by-product we obtain finite bases of the graded identities for sl2(K) graded by other groups such as ℤ2 × ℤ2 , and by the integers ℤ.

scholarly journals Central sets generated by uniformly recurrent words

2013 ◽  
Vol 35 (3) ◽  
pp. 714-736 ◽  
Author(s):  
MICHELANGELO BUCCI ◽  
SVETLANA PUZYNINA ◽  
LUCA Q. ZAMBONI
Keyword(s):  
Linear Equations ◽  
Combinatorial Structure ◽  
Combinatorics On Words ◽  
Combinatorial Properties ◽  
Finite Sums ◽  
Central Sets ◽  
The Rich ◽  
Numeration Systems ◽  
Sturmian Words ◽  
Homogeneous Linear

AbstractA subset $A$ of $ \mathbb{N} $ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $\mathop{({x}_{n} )}\nolimits_{n\in \mathbb{N} } $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of IP-sets possessing rich combinatorial properties: each central set contains arbitrarily long arithmetic progressions and solutions to all partition regular systems of homogeneous linear equations. In this paper we investigate central sets in the framework of combinatorics on words. Using various families of uniformly recurrent words, including Sturmian words, the Thue–Morse word and fixed points of weak mixing substitutions, we generate an assortment of central sets which reflect the rich combinatorial structure of the underlying words. The results in this paper rely on interactions between different areas of mathematics, some of which have not previously been directly linked. They include the general theory of combinatorics on words, abstract numeration systems, and the beautiful theory, developed by Hindman, Strauss and others, linking IP-sets and central sets to the algebraic/topological properties of the Stone-Čech compactification of $ \mathbb{N} $.

scholarly journals Some combinatorial properties of Sturmian words

1994 ◽  
Vol 136 (2) ◽  
pp. 361-385 ◽  
Author(s):  
Aldo de Luca ◽  
Filippo Mignosi
Keyword(s):  
Combinatorial Properties ◽  
Sturmian Words

ℤ2-graded identities of the Grassmann algebra over a finite field

2018 ◽  
Vol 28 (02) ◽  
pp. 291-307 ◽  
Author(s):  
Luís Felipe Gonçalves Fonseca
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Grassmann Algebra ◽  
Dimensional Vector ◽  
Polynomial Identities ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Graded Identities ◽  
Graded Polynomial Identities

Let [Formula: see text] be a finite field with the characteristic [Formula: see text] and let [Formula: see text] be the unitary Grassmann algebra generated by an infinite dimensional vector space [Formula: see text] over [Formula: see text]. In this paper, we determine a basis for [Formula: see text]-graded polynomial identities for any [Formula: see text]-grading such that its underlying vector space is homogeneous.

scholarly journals Weak polynomial identities and their applications

2021 ◽  
Vol 29 (2) ◽  
pp. 291-324
Author(s):  
Vesselin Drensky
Keyword(s):  
Associative Algebra ◽  
Finite Basis ◽  
Vector Subspace ◽  
Free Associative Algebra ◽  
Polynomial Identities ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Nonassociative Algebras ◽  
Survey Results ◽  
Algebra Over A Field

Abstract Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.

GRADED IDENTITIES FOR THE ALGEBRA OF n×n UPPER TRIANGULAR MATRICES OVER AN INFINITE FIELD

2003 ◽  
Vol 13 (05) ◽  
pp. 517-526 ◽  
Author(s):  
PLAMEN KOSHLUKOV ◽  
ANGELA VALENTI
Keyword(s):  
Infinite Field ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
The Ideal

We consider the algebra Un(K) of n×n upper triangular matrices over an infinite field K equipped with its usual ℤn-grading. We describe a basis of the ideal of the graded polynomial identities for this algebra.

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