On recognisable properties of associative algebras

Gateva-Ivanova, Tatiana; Latyshev, V. N.

doi:10.1016/s0747-7171(88)80054-3

Cited by 24 publications

(10 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This theorem was proved by T. Gateva-Ivanova [1] and [2] in the finitely presented case (see also [4,Proposition 15,Chapter 24] Proof. Let k be a natural number and w any subword of W with |u;| = (k + 2)|w|.…”

Section: Definitionmentioning

confidence: 94%

Radicals of monomial algebras

Kanel-Belov¹,

Gateva-Ivanova²

1996

First International Tainan-Moscow Algebra Workshop

View full text Add to dashboard Cite

It is proved that the Jacobson radical of a monomial algebra over a field is nil and a description of elements of the Jacobson radical is obtained.In what follows || £>|| will denote the cardinality of a set D, Z the ring of integers, F a field, Z = {zi, Z2, • • •, z n ) a finite set, (Z) the free semigroup on Z with unit element, and F{Z) the free F-algebra on Z. Given any subset Y c (Z), the factor algebra A = F(Z)/(Y) is called the monomial algebra defined by the set Y of words of (Z), where (7) is the ideal of F{Z) generated by Y. Note that(1)We denote by Nil (A) and J (A) the nil-radical and the Jacobson radical of A respectively. Next, we set X = {*, = z, + (Y) \ i = 1, 2,..., n}. Let,Xi 2 , ..., Xi m € X. The product u = Xi t Xj 2 ... Xj m e A is called a word in A, and we set m, if u ^ 0; |u| = 0, if w = 1 (i.e., m =0); [-1, if H = 0.In view of (1) the number |u| is well-defined. The number |«| is called the length of u. Let 1 < k < I < m and w = • • We shall call w a subword ofu and denote this fact by w An infinite sequence W = {Wj | w e W, j e Z} is said to be an infinite word in A. A subword u; of a product of the form WjWj + \... wj + / c is called a finite subword of W (and this fact is denoted by w W). An infinite word W is said to be nonzero if all of its finite subwords are nonzero. Definition 1.A nonzero infinite word W is called uniformly recurrent if for any integer k there exists an integer L = L(k) such that for any two finite subwords u and v of W with |u| = k and |u| = L we have thatThe following theorem is the main result of the present paper. Theorem 1. Let A be a finitely generated monomial algebra. Then J (A) = Nil(A). Furthermore, as a linear space, J (A) is spanned by all words which are not subwords of any uniformly recurrent word in A. 'Partially supported by the Russian Foundation for Fundamental Research.

show abstract

Section: Definitionmentioning

confidence: 94%

Radicals of monomial algebras

Kanel-Belov¹,

Gateva-Ivanova²

1996

First International Tainan-Moscow Algebra Workshop

View full text Add to dashboard Cite

show abstract

“…Then Π has polynomial growth if and only if each nonzero element of Π that is not nilpotent can be represented by a word whose primitive root labels some loop of the Ufnarovsky graph of the presentation. This follows from [6] where the syntactic structure of the Jacobson radical was described (also see this description in [24, Section 7.6, Theorem 3]) and the trivial fact that Π embeds in an associative monomial algebra with the same set of generators and defining relations (see also [13,Chapter 24]). PROOF.…”

Section: Bounded Heightmentioning

confidence: 99%

Growth of Finitely Presented Rees Quotients of Free Inverse Semigroups

Shneerson

Easdown

2011

Int. J. Algebra Comput.

View full text Add to dashboard Cite

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial growth if and only if it has bounded height. This occurs if and only if the set of nonzero reduced words has bounded Shirshov height and all nonzero reduced but not cyclically reduced words are nilpotent. This occurs also if and only if the set of nonzero geodesic words have bounded Shirshov height. We also give a simple sufficient graphical condition for polynomial growth, which is necessary when all zero relators are reduced. As a final application of our results, we give an inverse semigroup analogue of a classical result that characterises polynomial growth of finitely presented Rees quotients of free semigroups in terms of primitive words that label loops of the Ufnarovsky graph of the presentation. INTRODUCTIONIn [21] the authors initiated the study of growth of finitely presented Rees quotients of free inverse semigroups and the relationship with satisfiability of identities. In [5] they considered satisfiability of identities in signature with involution in the one relator case. Lau considered rationality of growth series [9] and degrees of growth [10] for semigroups from this class (see also [20] by the first author, in the wider context of Gelfand-Kirillov dimension and Rees quotients using infinitely many relators). In [21], growth was shown to be polynomial or exponential for semigroups from this class and an algorithmic criterion given to recognise which type of growth occurs [21, Section 3]. The arguments relied on a graphical technique, the ingredients of which are outlined in the next section, which is a modification of an idea due to Ufnarovsky [23] [24] (see also [13, Chapter 24]). In Theorem 3.3 below a simplified graphical condition is given which is always sufficient for polynomial growth, and necessary when all zero relators are reduced. Shirshov [18] (see also [16, Chapter 4], [22]) introduced the notion of height in his celebrated study of rings which satisfy a polynomial identity. In Theorem 4.3 in the final section we give simple criteria for polynomial growth for semigroups from our class in terms of height, nilpotency and geodesic words. Some of the main results of the final section also appear in Chapter 4 of Lau's doctoral thesis [11], but with different proofs. The final result, Theorem 4.4, is an application of our results to yield an inverse semigroup analogue of a classical result characterising polynomial growth 2 of finitely presented Rees quotients of free semigroups in terms of primitive words that label loops of the Ufnarovsky graph. PRELIMINARIESWe assume familiarity with the basic definitions and elementary results from the theory of semigroups, which can be found in any of [3], [7], [8] or [14]. Let S be a semigroup generated by a finite subset X . Recall that the length l(t) of an element t ∈ S (with respect to X ) is the least number of factors in all representations of t as a product of elements of X , andis called the growth function of S . We say that S has polynomial growth...

show abstract

“…ExampIe: k(x, y 1 xy = y2 = x4 = 0, x3 = yx) is nondegenerate, but k(z 1 z2 = az) and L (z 1 z2 = 1) are not. Remark 1: The dimension finiteness implies that condition (2) in Definition 1 can be replaced by condition (2') which says that there is not a nonzero a E k such that vu = au. Remark 2: A monomial algebra is a non-degenerate binomial algebra.…”

Section: The Word Posetsmentioning

confidence: 99%

“…Let us call such a path a linked path. Here p(A) z p(B) as word posets w there exists a poset isomorphism u: p(A) + p(B) such that (1) the restriction of 0 to X gives a bijection from X into Y (therefore #X = fly), and (2) if uu is in p(A), then a( is in p(B) and a(u)o(v) = u(uv).…”

Section: The Word Posetsmentioning

confidence: 99%

See 1 more Smart Citation

On the isomorphism problem for finite-dimensional binomial algebras

Shirayanagi

1990

Proceedings of the International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

Binomial algebras are finitely presented algebras defined by monomials or binomials. Given two binomial algebras, one important problem is to decide whether or not they are isomorphic as algebras. We study an algorithm for solving this problem, when both algebras are finite-dimensional over a field. In particular, when they are monomial algebras {i.e. binomial algebras defined by monomials only), the problem has already been completely solved by the presentation uniqueness.In this paper, we provide some necessary conditions in terms of partiahy ordered sets for two certain binomial algebras to be isomorphic. In other words, invariants of the binomial algebras are presented. These conditions together serve as an effective procedure for solving the isomorphism problem.

show abstract

On recognisable properties of associative algebras

Cited by 24 publications

References 4 publications

Radicals of monomial algebras

Radicals of monomial algebras

Growth of Finitely Presented Rees Quotients of Free Inverse Semigroups

On the isomorphism problem for finite-dimensional binomial algebras

Contact Info

Product

Resources

About