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Essential pure monomorphisms of sheaves of group actions

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Abstract

Nowadays purity plays a role in at least four branches of mathematics: Module Theory, Theory of Acts over semigroups, Model Theory, and Category Theory. Adámek and Rosciký have studied these notions categorically, and Rothmaler model-theoretically. Some authors including Banaschewski, Gould, and Normak have studied purity on G-acts, acts over a monoid or a group G.

In this paper we take both the group G and the set A in the definition of a G-act to be sheaves (multigroups and multisets) over a topological space X to get the category G-ShX of G-sheaves, and mainly study different types of essentiality with respect to the different classes ℳ of pure monomorphisms in G-ShX. We give some necessary and sometimes sufficient conditions for the different types of essentiality.

We will also see that these notions of essentiality are local, in the sense that an ℳ-monomorphism f is ℳ-essential if and only if there exists a cover of disjoint open sets of X such that the restriction of f to every component is essential.

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Authors and Affiliations

  1. Department of Mathematics, Shahid Beheshti University, G.C., Tehran, 19839, Iran

    M. M. Ebrahimi & M. Haddadi

Authors
  1. M. M. Ebrahimi
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  2. M. Haddadi
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Correspondence to M. M. Ebrahimi.

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Communicated by László Márki.

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Ebrahimi, M.M., Haddadi, M. Essential pure monomorphisms of sheaves of group actions. Semigroup Forum 80, 440–452 (2010). https://doi.org/10.1007/s00233-010-9212-7

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  • DOI: https://doi.org/10.1007/s00233-010-9212-7

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