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Jonathan Brundan - One of the best experts on this subject based on the ideXlab platform.
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A Basis Theorem for the degenerate affine oriented Brauer-Clifford supercategory
2019Co-Authors: Jonathan Brundan, Jonathan Comes, Jonathan R. KujawaAbstract:We introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are Basis Theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.
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A Basis Theorem for the affine oriented Brauer category and its cyclotomic quotients
2017Co-Authors: Jonathan Brundan, Jonathan Comes, David A. Nash, Andrew ReynoldsAbstract:The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category (= the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropriate relations. In this article, we prove a Basis Theorem for the morphism spaces in this category, as well as for all of its cyclotomic quotients.
Sophie Huczynska - One of the best experts on this subject based on the ideXlab platform.
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The strong primitive normal Basis Theorem
2010Co-Authors: Stephen D. Cohen, Sophie HuczynskaAbstract:An element α of the extension E of degree n over the finite field F = GF(q) is called free over F if {α, α q , . . . , α q n 1 } is a (normal) Basis of E/F. The primitive normal Basis Theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F, there exists an element α ∈ E such that α is simultaneously primitive (i.e., generates the multiplicative group of E) and free over F. In this paper we prove the following strengthening of this Theorem: aside from four specific extensions E/F, there exists an element α ∈ E such that both α and α −1 are simultaneously primitive and free over F. AMS classification: Primary 11T30; Secondary 11T06, 12E20
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The Strong Primitive Normal Basis Theorem
2006Co-Authors: Stephen D. Cohen, Sophie HuczynskaAbstract:An element w of the extension E of degree n over the finite field F=GF(q) is called free over F if {w, w^q,...,w^{q^{n-1}}} is a (normal) Basis of E/F. The Primitive Normal Basis Theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F, there exists an element w in E such that w is simultaneously primitive (i.e., generates the multiplicative group of E) and free over F. In this paper we prove the following strengthening of this Theorem: aside from five specific extensions E/F, there exists an element w in E such that both w and w^{-1} are simultaneously primitive and free over F.
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the primitive normal Basis Theorem without a computer
2003Co-Authors: Stephen D. Cohen, Sophie HuczynskaAbstract:Given $q$ , a power of a prime $p$ , denote by $F$ the finite field ${\rm GF}(q)$ of order $q$ , and, for a given positive integer $n$ , by $E$ its extension ${\rm GF}(q^n)$ of degree $n$ . A primitive element of $E$ is a generator of the cyclic group $E^\ast$ . Additively too, the extension $E$ is cyclic when viewed as an $FG$ -module, $G$ being the Galois group of $E$ over $F$ . The classical form of this result – the normal Basis Theorem – is that there exists an element $\alpha \in E$ (an additive generator) whose conjugates $\{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ form a Basis of $E$ over $F; \alpha$ is a free element of $E$ over $F$ , and a Basis like this is a normal Basis over $F$ . The core result linking additive and multiplicative structure is that there exists $\alpha \in E$ , simultaneously primitive and free over $F$ . This yields a primitive normal Basis over $F$ , all of whose members are primitive and free. Existence of such a Basis for every extension was demonstrated by Lenstra and Schoof [ 5 ] (completing work by Carlitz [ 1 , 2 ] and Davenport [ 4 ]).
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the primitive normal Basis Theorem without a computer
2003Co-Authors: Stephen D. Cohen, Sophie HuczynskaAbstract:Given , a power of a prime , denote by the finite field of order , and, for a given positive integer , by its extension of degree . A primitive element of is a generator of the cyclic group . Additively too, the extension is cyclic when viewed as an -module, being the Galois group of over . The classical form of this result – the normal Basis Theorem – is that there exists an element (an additive generator) whose conjugates form a Basis of over is a free element of over , and a Basis like this is a normal Basis over . The core result linking additive and multiplicative structure is that there exists , simultaneously primitive and free over . This yields a primitive normal Basis over , all of whose members are primitive and free. Existence of such a Basis for every extension was demonstrated by Lenstra and Schoof [5] (completing work by Carlitz [1, 2] and Davenport [4]).
Jonathan Comes - One of the best experts on this subject based on the ideXlab platform.
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A Basis Theorem for the degenerate affine oriented Brauer-Clifford supercategory
2019Co-Authors: Jonathan Brundan, Jonathan Comes, Jonathan R. KujawaAbstract:We introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are Basis Theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.
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A Basis Theorem for the affine oriented Brauer category and its cyclotomic quotients
2017Co-Authors: Jonathan Brundan, Jonathan Comes, David A. Nash, Andrew ReynoldsAbstract:The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category (= the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropriate relations. In this article, we prove a Basis Theorem for the morphism spaces in this category, as well as for all of its cyclotomic quotients.
Andrew Reynolds - One of the best experts on this subject based on the ideXlab platform.
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A Basis Theorem for the affine oriented Brauer category and its cyclotomic quotients
2017Co-Authors: Jonathan Brundan, Jonathan Comes, David A. Nash, Andrew ReynoldsAbstract:The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category (= the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropriate relations. In this article, we prove a Basis Theorem for the morphism spaces in this category, as well as for all of its cyclotomic quotients.
Ross Willard - One of the best experts on this subject based on the ideXlab platform.
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Congruence meet-semidistributive locally finite varieties and a finite Basis Theorem
2018Co-Authors: George F. Mcnulty, Ross WillardAbstract:We provide several conditions that, among locally finite varieties, characterize congruence meet-semidistributivity and we use these conditions to give a new proof of a finite Basis Theorem published by Baker, McNulty, and Wang in 2004. This finite Basis Theorem extends Willard’s Finite Basis Theorem.
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A finite Basis Theorem for difference-term varieties with a finite residual bound
2015Co-Authors: Keith A. Kearnes, Ágnes Szendrei, Ross WillardAbstract:We prove that if V is a variety of algebras (i.e., an equationally axiomatizable class of algebraic structures) in a finite language, V has a difference term, and V has a finite residual bound, then V is finitely axiomatizable. This provides a common generalization of R. McKenzie’s finite Basis Theorem for congruence modular varieties with a finite residual bound, and R. Willard’s finite Basis Theorem for congruence meet-semidistributive varieties with a finite residual bound. This paper is a contribution to an old problem in logic from the schools of A. Tarski and A. Maltsev: which finite algebraic structures A (algebras for short) have a finite Basis for their identities? Equivalently, for which finite algebras A is the variety V(A) (the smallest equational class containing A) finitely axiomatizable? On the one hand, every finite group [38], finite ring [22, 28], finite commutative semigroup [40], finite lattice [33], or two-element algebra in a finite language [30] is known to be finitely based. On the other hand, the list of finite algebras which are not finitely based includes, in addition to pathological examples (e.g., [31, 37]), some finite semigroups [40], some finite non-associative K-algebras [41, 29, 12], and even a finite group with one non-identity element named by a constant [5]. In 1996 R. McKenzie [36] proved that the problem of determining whether a finite algebra is finitely based is undecidable, settling Tarski’s finite Basis problem. The evidence suggests that a full classification of finitely based finite algebras is beyond reach. However, there are some remarkable partial results. In particular, in the early 1970s K. Baker [1, 2] proved the following: if A is a finite algebra in a finite language and V(A) is congruence distributive (i.e., for every B ∈ V(A), the lattice of congruence relations of B is a distributive lattice), then A is finitely based. Two important ingredients in the proof were provided by B. Jonsson [13]: (1) a characterization, in terms of identities, of the condition that a variety be congruence distributive, and (2) a proof that if A is finite and V(A) is congruence distributive, then every subdirectly Date: July 8, 2014. 2010 Mathematics Subject Classification. Primary 03C05; Secondary 08B05.
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a finite Basis Theorem for residually finite congruence meet semidistributive varieties
2000Co-Authors: Ross WillardAbstract:We derive a Mal'cev condition for congruence meet-semidistributivity and then use it to prove two Theorems. Theorem A: if a variety in a finite language is congruence meet-semidistributive and residually less than some finite cardinal, then it is finitely based. Theorem B: there is an algorithm which, given m < ω and a finite algebra in a finite language, determines whether the variety generated by the algebra is congruence meet-semidistributive and residually less than m.
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a finite Basis Theorem for residually finite congruence meet semidistributive varieties
1997Co-Authors: Ross WillardAbstract:We derive a Mal'cev condition for congruence meet-semidistributivity and then use it to prove two Theorems. Theorem A: if a variety in a finite language is congruence meet-semidistributive and residually less than some finite cardinal, then it is finitely based. Theorem B: there is an algorithm which, given m
algebra in a finite language, determines whether the variety generated by the algebra is congruence meet-semidistributive and residually less then m.