Noetherian Semigroup Algebras

Jespers, Eric; Okniński, Jan

doi:10.1007/1-4020-5810-1

Cited by 40 publications

(30 citation statements)

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“…Notice that G(X, r) = σ 1 , σ 2 , σ 3 , σ 4 is isomorphic to the dihedral group of order 8. This is an example of an indecomposable and irretractable solution, see [24] or Example 8.2.14 in [25]; actually the first known example of a solution whose structure group is not a poly-Z group. It is clear that G(X, r) acts transitively on X and X 1 = {1, 4} and X 2 = {2, 3} form imprimitivity blocks for the action of the group G(X, r) on X.…”

Section: Indecomposable and Simple Solutionsmentioning

confidence: 99%

“…(25)By(14),σ u σ v (l) = σ d k,0 (v) σ d −1 k,0 (u) (l), for all u, v, k, l ∈ Z/(p). In particular, σ ν(u) σ ν(v) (η(l)) = σ d k,0 (ν(v)) σ d −1 k,0 (ν(u)) (η(l)).Hence, applying this and condition(25) several times, we getη(t(tl + v) + u) = σ ν(u) (η(tl + v)) = σ ν(u) (σ ν(v) (η(l))) = σ d k,0 (ν(v)) σ d −1 k,0 (ν(u)) (η(l)) = σ d k,0 (ν(v)) (η(tl + ν −1 (d −1 k,0 (ν(u))))) = η(ν −1 (d k,0 (ν(v))) + t(tl + ν −1 (d −1 k,0 (ν(u))))),and thus tv+ u = ν −1 (d k,0 (ν(v))) + tν −1 (d −1 k,0 (ν(u))), for all u, v, k ∈ Z/(p). In particular, for u = 0 we getν −1 (d k,0 (ν(v))) = t(v − ν −1 (d −1 k,0 (ν(0)))).…”

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confidence: 91%

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Constructing finite simple solutions of the Yang-Baxter equation

Cedó¹,

Okniński²

2020

Preprint

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We study involutive non-degenerate set-theoretic solutions (X, r) of the Yang-Baxter equation on a finite set X. The emphasis is on the case where (X, r) is indecomposable, so the associated permutation group G(X, r) acts transitively on X. One of the major problems is to determine how such solutions are built from the imprimitivity blocks; and also how to characterize these blocks. We focus on the case of so called simple solutions, which are of key importance. Several infinite families of such solutions are constructed for the first time. In particular, a broad class of simple solutions of order p 2 , for any prime p, is completely characterized.

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Section: Indecomposable and Simple Solutionsmentioning

confidence: 99%

mentioning

confidence: 91%

Constructing finite simple solutions of the Yang-Baxter equation

Cedó¹,

Okniński²

2020

Preprint

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“…) and those intersecting S n nontrivially. As in the case of other classes of semigroup algebras, one may expect that the latter play an important role in the study of the properties of the algebra, see [16]. The first step is to look at the minimal prime ideals of S n and their impact on the structure of K[S n ].…”

Section: Because This Map Can Be Extended To a Bijectionmentioning

confidence: 99%

“…Recently there has been extensive interest in some finitely presented algebras A defined by homogeneous semigroup relations, that is, relations of the form w = v, where w and v are words of the same length in a generating set of the algebra. A particular intriguing class (defined by homogeneous relations of degree 2) are the algebras yielding set theoretic solutions of the Yang-Baxter equation, [8,11,12,13,15,16,24]. Such algebras do not only have nice ring theoretic properties but they also lead to exciting groups G and monoids S defined by the same presentation.…”

Section: Introductionmentioning

confidence: 99%

Finitely presented algebras and groups defined by permutation relations

Cedó

Jespers

Okniński

2010

Journal of Pure and Applied Algebra

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Communicated by M. Sapir MSC: 16S15 16S36 20M05 20M25 a b s t r a c tThe class of finitely presented algebras over a field K with a set of generators a 1 , . . . , a n and defined by homogeneous relations of the form a 1 a 2 · · · a n = a σ (1) a σ (2) · · · a σ (n) , where σ runs through a subset H of the symmetric group Sym n of degree n, is introduced. The emphasis is on the case of a cyclic subgroup H of Sym n of order n. A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups H of Sym n are proposed.

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“…Normalizing Krull Monoids. A semigroup H is normalizing if aH = Ha for all a ∈ H. Normalizing Krull Monoids occur when studying Noetherian semigroup algebras [82] and are Transfer Krull by [49,Theorems 4.13 an 6.5].…”

Section: Introductionmentioning

confidence: 99%

The Characterization of Finite Elasticities

Grynkiewicz¹

2020

Preprint

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Our main motivating goal is the study of factorization in Krull Domains H with finitely generated class group G. While factorization into irreducibles, called atoms, generally fails to be unique, there are various measures of how badly this can fail. One of the most important is the elasticity ρ(H) = lim k→∞ ρ k (H)/k, where ρ k (H) is the maximal number of atoms in any re-factorization of a product of k atoms. Having finite elasticity is a key indicator that factorization, while not unique, is not completely wild. The elasticities, as well as many other arithmetic invariants, are the same as those of an associated combinatorial monoid B(G0) of zero-sum sequences, where G0 ⊆ G are the classes containing height one primes.We characterize when finite elasticity holds for any Krull Domain with finitely generated class group G. Indeed, our results are valid for the more general class of Transfer Krull Monoids (over a subset G0 of a finitely generated abelian group G). Moreover, we show there is a minimal s ≤ (d + 1)m, where d is the torsion free rank and m is the exponent of the torsion subgroup, such that ρsOur characterization is in terms of a simple combinatorial obstruction to infinite elasticity: there existing a subset G ⋄ 0 ⊆ G0 and global bound N such that there are no nontrivial zero-sum sequences with terms from G ⋄ 0 , and every minimal zero-sum sequence has at most N terms from G0 \ G ⋄ 0 . We give an explicit description of G ⋄ 0 in terms of the Convex Geometry of G0 modulo the torsion subgroup GT ≤ G, and show that finite elasticity is equivalent to there being no positive R-linear combination of the elements of this explicitly defined subset G ⋄ 0 equal to 0 modulo GT . Additionally, we use our results to show finite elasticity implies the set of distances ∆(H), the catenary degree c(H) (for Krull Monoids) and a weakened form of the tame degree (for Krull Monoids) are all also finite, and that the Structure Theorem for Unions holds-four of the most commonly used measurements of structured factorization, after the elasticity.Our results for factorization in Transfer Krull Monoids are accomplished by developing an extensive theory in Convex Geometry generalizing positive bases. The convex cone generated byPositive bases were first introduced and studied in the mid 20th century, and the structural work initiated by Reay led to a simplified proof and strengthening of Bonnice and Klee's celebrated generalization of Carathéordory's Theorem. They have since found increasing importance in areas of applied mathematics. We show that the structural result of Reay can be extended to special types of complete simplicial fans, which we term Reay systems. We extend these results to a general theory dealing with infinite sequences of Reay systems, as well as their limit structures. The latter, while more complex, avoid the introduction of linear dependencies into the limit structure that were not originally present, circumventing the general obstacle that a limit of linearly independent sets can degenerate into linear...

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Noetherian Semigroup Algebras

Cited by 40 publications

References 98 publications

Constructing finite simple solutions of the Yang-Baxter equation

Constructing finite simple solutions of the Yang-Baxter equation

Finitely presented algebras and groups defined by permutation relations

The Characterization of Finite Elasticities

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