Gröbner bases for quadratic algebras of skew type

2012 ◽  
Vol 55 (2) ◽  
pp. 387-401 ◽  
Author(s):  
Ferran Cedó ◽  
Jan Okniński
Keyword(s):  
Regular Language ◽  
Normal Forms ◽  
Free Monoid ◽  
Lexicographic Order ◽  
Semigroup Algebra ◽  
Baxter Equation ◽  
Finitely Generated ◽  
Quadratic Algebras ◽  
Defining Relations ◽  
Finite Module

AbstractNon-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and $\binom{n}{2}$ quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

scholarly journals NORMAL DOMAINS WITH MONOMIAL PRESENTATIONS

2009 ◽  
Vol 19 (03) ◽  
pp. 287-303 ◽  
Author(s):  
ISABEL GOFFA ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Commutative Algebra ◽  
Class Group ◽  
Semigroup Algebra ◽  
Closed Domain ◽  
Finitely Generated ◽  
Defining Relations ◽  
Integrally Closed ◽  
Algebra Over A Field

Let A be a finitely generated commutative algebra over a field K with a presentation A = K 〈X1,…, Xn | R〉, where R is a set of monomial relations in the generators X1,…, Xn. So A = K[S], the semigroup algebra of the monoid S = 〈X1,…, Xn | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

A presentation of a finitely generated submonoid of invertible endomorphisms of the free monoid

2015 ◽  
Vol 93 (3) ◽  
pp. 444-458
Author(s):  
Christian Choffrut ◽  
Štěpán Holub
Keyword(s):  
Free Monoid ◽  
Finitely Generated

scholarly journals Markov semigroups, monoids and groups

2014 ◽  
Vol 24 (05) ◽  
pp. 609-653 ◽  
Author(s):  
Alan J. Cain ◽  
Victor Maltcev
Keyword(s):  
Finite Index ◽  
Regular Language ◽  
Minimal Length ◽  
Direct Products ◽  
Markov Semigroups ◽  
Finitely Generated ◽  
Free Products ◽  
Commutative Semigroups ◽  
Generating Set ◽  
Polycyclic Groups

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.

Affine algebras of Gelfand-Kirillov dimension one are PI

1985 ◽  
Vol 97 (3) ◽  
pp. 407-414 ◽  
Author(s):  
L. W. Small ◽  
J. T. Stafford ◽  
R. B. Warfield
Keyword(s):  
Polynomial Identity ◽  
Prime Radical ◽  
Finitely Generated ◽  
Finite Module ◽  
Affine Algebras ◽  
Dimension One ◽  
Kirillov Dimension ◽  
Algebra Over A Field

The aim of this paper is to prove:Theorem.Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

scholarly journals Gröbner-Shirshov Bases Theory for Trialgebras

2021 ◽  
Author(s):  
Juwei Huang ◽  
Yuqun Chen
Keyword(s):  
Normal Forms ◽  
Finitely Generated ◽  
Kirillov Dimension

We establish a Gröbner-Shirshov bases theory for trialgebras and show that every ideal of a free trialgebra has a unique reduced Gröbner-Shirshov basis. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand-Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated respectively.

scholarly journals Abstract commensurators of surface groups

2019 ◽  
pp. 1-16
Author(s):  
Khalid Bou-Rabee ◽  
Daniel Studenmund
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Computational Methods ◽  
Normal Forms ◽  
Computer Assisted ◽  
Surface Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Hnn Extensions ◽  
Concrete Objects

Let [Formula: see text] be the fundamental group of a surface of finite type and [Formula: see text] be its abstract commensurator. Then [Formula: see text] contains the solvable Baumslag–Solitar groups [Formula: see text] for any [Formula: see text]. Moreover, the Baumslag–Solitar group [Formula: see text] has an image in [Formula: see text] that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of [Formula: see text] are concrete objects amenable to computational methods. For example, we give a proof that [Formula: see text] is not residually finite without the use of normal forms of HNN extensions.

scholarly journals Poincaré series of character varieties for nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 419-440 ◽  
Author(s):  
Mentor Stafa
Keyword(s):  
Lie Group ◽  
Free Monoid ◽  
Nilpotent Groups ◽  
Poincaré Series ◽  
Character Variety ◽  
Finitely Generated ◽  
Trivial Representation ◽  
Poincare Series ◽  
Character Varieties ◽  
Path Component

Abstract For any compact, connected Lie group G and any finitely generated nilpotent group Γ, we determine the cohomology of the path component of the trivial representation of the group character variety (representation space) {{\rm Rep}(\Gamma,G)_{1}} , with coefficients in a field {{\mathbb{F}}} with characteristic 0 or relatively prime to the order of the Weyl group W. We give explicit formulas for the Poincaré series. In addition, we study G-equivariant stable decompositions of subspaces {{\rm X}(q,G)} of the free monoid {J(G)} generated by the Lie group G, obtained from representations of finitely generated free nilpotent groups.

AN APPLICATION OF FIRST-ORDER LOGIC TO THE STUDY OF RECOGNIZABLE LANGUAGES

2004 ◽  
Vol 14 (05n06) ◽  
pp. 785-799 ◽  
Author(s):  
PEDRO V. SILVA
Keyword(s):  
Finite Rank ◽  
Free Monoid ◽  
Order Logic ◽  
First Order Logic ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
First Order ◽  
Free Inverse Monoid

A variation of first-order logic with variables for exponents is developed to solve some problems in the setting of recognizable languages on the free monoid, accommodating operators such as product, bounded shuffle and reversion. Restricting the operators to powers and product, analogous results are obtained for recognizable languages of an arbitrary finitely generated monoid M, in particular for a free inverse monoid of finite rank. As a consequence, it is shown to be decidable whether or not a recognizable subset of M is pure or p-pure.

QUADRATIC ALGEBRAS OF SKEW TYPE SATISFYING THE CYCLIC CONDITION

2004 ◽  
Vol 14 (04) ◽  
pp. 479-498 ◽  
Author(s):  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Baxter Equation ◽  
Important Class ◽  
Polynomial Algebras ◽  
Quadratic Algebras ◽  
Cyclic Condition ◽  
Special Case ◽  
Special Classes

We consider algebras over a field with generators x1,x2,…,xn subject to [Formula: see text] square-free relations xixj=xkxl in which every product xpxq, p≠q, appears in one of the relations. The work of Gateva-Ivanova and Van den Bergh, motivated in particular by the study of set theoretic solutions of the Yang–Baxter equation, provided an important class of such algebras. In this special case the presentation satisfies the so-called cyclic condition that became an essential combinatorial tool in proving that these algebras share many strong ring theoretic properties of polynomial algebras in commuting variables. In this paper we describe the structure of algebras on four generators satisfying the cyclic condition. The emphasis is on some new unexpected features, not present in the motivating special classes.

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