Potent preservers of incidence algebras

Garcés, Jorge J.; Khrypchenko, Mykola

doi:10.1016/j.laa.2021.11.020

Cited by 7 publications

(5 citation statements)

References 30 publications

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“…On the other hand, by (13) and Lemma 3.1 (vi)a we have ϕ(e ux )e λ(z) = 0 for all z ∈ {u, x}, so combining this with ( 14…”

Section: Thus In What Follows We Assumementioning

confidence: 94%

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Linear maps preserving products equal to primitive idempotents of an incidence algebra

Garcés¹,

Khrypchenko²

2022

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Let A, B be algebras and a ∈ A, b ∈ B a fixed pair of elements. We say that a map ϕ : A → B preserves products equal to a and b if for all a 1 , a 2 ∈ A the equality a 1 a 2 = a implies ϕ(a 1 )ϕ(a 2 ) = b. In this paper we study bijective linear maps ϕ : I(X, F ) → I(X, F ) preserving products equal to primitive idempotents of I(X, F ), where I(X, F ) is the incidence algebra of a finite connected poset X over a field F . We fully characterize the situation, when such a map ϕ exists, and whenever it does, ϕ is either an automorphism of I(X, F ) or the negative of an automorphism of I(X, F ).

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“…On the other hand, by (13) and Lemma 3.1 (vi)a we have ϕ(e ux )e λ(z) = 0 for all z ∈ {u, x}, so combining this with ( 14…”

Section: Thus In What Follows We Assumementioning

confidence: 94%

“…Since {ψ(e z )} z∈X\{x} is a set of orthogonal idempotents and |X \ {x}| = |X \ {y}|, there exists a bijection λ ′ : X \ {x} → X \ {y} such that ψ(e z ) D = e λ ′ (z) . Hence, by [13,Lemma 5.4] there is β ′ ∈ I(X \ {y}, F ) * such that β ′ ψ(e z )β ′−1 = e λ ′ (z) for all z ∈ X \ {x}. Then…”

Section: Thus In What Follows We Assumementioning

confidence: 95%

Linear maps preserving products equal to primitive idempotents of an incidence algebra

Garcés¹,

Khrypchenko²

2022

Preprint

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“…The study of algebraic mappings on incidence algebras was initiated by Stanley [237]. Since then, automorphisms, involutions, and derivations (and their generalizations) on incidence algebras have been actively investigated, see [100,103,218] and the references therein.…”

Section: Higher Derivations Of Incidence Algebrasmentioning

confidence: 99%

Non-associative algebraic structures: classification and structure

Kaygorodov

2023

Communications in Mathematics

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“…If [S α (f ), S α (g)] = 0, then [f, g](x, y) = 0 for all x < y with l(x, y) > 1 by (11). Therefore, (11) (10). This implies (8).…”

mentioning

confidence: 93%

“…Let e xy ∈ B and z ∈ {x, y}. Then [e z , e xy ] = 0, so (6) follows from (10) and the commutativity preserving property of S α . Now let e xy ∈ B with l(x, y) > 1.…”

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confidence: 99%

Commutativity preservers of incidence algebras

Fornaroli¹,

Khrypchenko²,

Santulo³

2022

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Let I(X, K) be the incidence algebra of a finite connected poset X over a field K and D(X, K) its subalgebra consisting of diagonal elements. We describe the bijective linear maps ϕ : I(X, K) → I(X, K) that strongly preserve the commutativity and satisfy ϕ(D(X, K)) = D(X, K). We prove that such a map ϕ is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple (θ, σ, c, κ) of simpler maps θ, σ, c and a sequence κ of elements of K.

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Potent preservers of incidence algebras

Cited by 7 publications

References 30 publications

Linear maps preserving products equal to primitive idempotents of an incidence algebra

Linear maps preserving products equal to primitive idempotents of an incidence algebra

Non-associative algebraic structures: classification and structure

Commutativity preservers of incidence algebras

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