Automorphisms of the endomorphism semigroup of a polynomial algebra

Belov-Kanel, Alexei; Lipyanski, Ruvim

doi:10.1016/j.jalgebra.2011.01.020

Cited by 9 publications

(7 citation statements)

References 14 publications

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“…Remark. We are grateful to Robin Chapman for independent confirmation of the values for PΓL (2,32) in Table 3. Table 3.…”

Section: Belowmentioning

confidence: 89%

“…Table 7 gives the number of G-orbits on the 14-partitions of various types, and the numbers with each of the three possible stabilisers. For the group G = PSL(2, 2 p ), with p prime and p > 3, the situation is much simpler: no (2 p − 2) partition can be fixed by an element outside G, and so every orbit has stabiliser G. (This also shows, for example, that the numbers of orbits for PSL (2,32) are five times those for PΓL (2,32) given in Table 6. )…”

Section: Orbits Of Normalizersmentioning

confidence: 97%

“…(iv) AΓL (1,8), M 11 (k = 4), M 11 (degree 12, k = 3), M 12 (k = 5), 2 4 : A 7 , M 22 : 2 (k = 3), M 23 (k = 4), M 24 (k = 5), and AΓL (1,32) (1,8), PGL (2,8), PGL (2,9), M 10 , PSL (2,11), M 22 , PXL (2,25), or PXL (2,49), with k = 3 and λ = (4, 1, . .…”

Section: Introductionmentioning

confidence: 99%

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Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Araújo¹,

Bentz

Cameron

2018

Trans. Amer. Math. Soc.

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Let X be a finite set such that |X| = n, and let k < n/2. A group is k-homogeneous if it has only one orbit on the sets of size k. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate k-homogeneous groups (for k ≥ 1); in particular we show that 2-homogeneous groups are 2-generated. We also describe the orbits of k-homogenous groups on partitions with n − k parts, classify the 3-homogeneous groups G whose orbits on (n − 3)-partitions are invariant under the normalizer of G in S n , and describe the normalizers of 2-homogeneous groups in the symmetric group. Then these results are applied to extract information about transformation semigroups with given group of units, namely to prove results on their automorphisms and on the minimum number of generators. The paper finishes with some problems on permutation groups, transformation semigroups and computational algebra.

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“…Remark. We are grateful to Robin Chapman for independent confirmation of the values for PΓL (2,32) in Table 3. Table 3.…”

Section: Belowmentioning

confidence: 89%

Section: Orbits Of Normalizersmentioning

confidence: 97%

See 1 more Smart Citation

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Araújo¹,

Bentz

Cameron

2018

Trans. Amer. Math. Soc.

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“…In the same spirit, the description of an endomorphism semigroup of the ring of commutative polynomials A is given by Belov and Lipyanski in Reference [100]: Theorem 7. Every automorphism of the group Aut(End(A)) is semi-inner.…”

Section: Plotkin's Problem: Automorphisms Of Endomorphism Semigroups mentioning

confidence: 99%

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

et al. 2020

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This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given.

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“…On the other hand, many instances of this problem have been solved for other classes of algebras [58,59,63]. (See also [7,10,12,17,18,19,20,29,30,31,53]. )…”

Section: Problemsmentioning

confidence: 99%

The largest subsemilattices of the endomorphism monoid of an independence algebra

Araújo

Bentz

Konieczny

2014

Linear Algebra and its Applications

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An algebra A is said to be an independence algebra if it is a matroid algebra and every map α : X → A, defined on a basis X of A, can be extended to an endomorphism of A. These algebras are particularly well behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics such as model theory, group theory, and semigroup theory.It is well known that matroid algebras have a well defined notion of dimension. Let A be any independence algebra of finite dimension n, with at least two elements. Denote by End(A) the monoid of endomorphisms of A. We prove that a largest subsemilattice of End(A) has either 2 n−1 elements (if the clone of A does not contain any constant operations) or 2 n elements (if the clone of A contains constant operations). As corollaries, we obtain formulas for the size of the largest subsemilattices of: some variants of the monoid of linear operators of a finite-dimensional vector space, the monoid of full transformations on a finite set X, the monoid of partial transformations on X, the monoid of endomorphisms of a free G-set with a finite set of free generators, among others.The paper ends with a relatively large number of problems that might attract attention of experts in linear algebra, ring theory, extremal combinatorics, group theory, semigroup theory, universal algebraic geometry, and universal algebra.2010 Mathematics Subject Classification. 08A35, 20M20.

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Automorphisms of the endomorphism semigroup of a polynomial algebra

Cited by 9 publications

References 14 publications

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

The largest subsemilattices of the endomorphism monoid of an independence algebra

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