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Commutativity Preservers of Incidence Algebras

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Abstract

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 \(\varphi :I(X,K)\rightarrow I(X,K)\) that strongly preserve the commutativity and satisfy \(\varphi (D(X,K))=D(X,K)\). We prove that such a map \(\varphi \) is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple \((\theta ,\sigma ,c,\kappa )\) of simpler maps \(\theta \), \(\sigma \), c and a sequence \(\kappa \) of elements of K.

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Funding

The second author was partially supported by CNPq (process: 404649/2018-1).

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Estadual de Maringá, Maringá, PR, CEP, 87020–900, Brazil

    Érica Z. Fornaroli & Ednei A. Santulo Jr

  2. Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Reitor João David Ferreira Lima, Florianópolis, SC, CEP, 88040–900, Brazil

    Mykola Khrypchenko

Authors
  1. Érica Z. Fornaroli
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  2. Mykola Khrypchenko
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  3. Ednei A. Santulo Jr
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All authors wrote the main manuscript text.

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Correspondence to Mykola Khrypchenko.

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Presented by: Cristian Lenart

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Fornaroli, É.Z., Khrypchenko, M. & Santulo, E.A. Commutativity Preservers of Incidence Algebras. Algebr Represent Theor (2024). https://doi.org/10.1007/s10468-024-10265-x

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