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The small inductive dimension of finite lattices through matrices

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Abstract

Dimensions of partially ordered sets, lattices and frames have attracted the interest of various researches (see for example Sancho de Salas and Sancho de Salas 1991; Vinokurov 1966; Dushnik and Miller 1941; Hegde and Jain 2007; Trotter 1975; Boyadzhiev et al. 2018; Dube et al. 2015, 2017; Georgiou et al. 2016; Hai-feng et al. 2017; Brijlall and Baboolal 2008, 2010; Boyadzhiev et al. 2019). For example, the order, Krull, covering dimension, and the quasi covering dimension are some of these dimensions that have been studied extensively. Especially, finite partially ordered sets and finite lattices are the main axes of this Dimension Theory, investigating new results and characterizations for their dimensions. The matrix algebra plays an essential role in these studies, considering incidence and order matrices. Also, based on these researches, algorithms which compute these dimensions have been investigated. Since the chapter on dimensions of partially ordered sets continues to attract the interest, the small inductive dimension is a new notion that has been defined firstly for regular frames in Brijlall and Baboolal (2008). In this paper, we study the small inductive dimension for finite lattices using matrices. For that, we study the meaning of the pseudocomplement and the lattice \({\uparrow }x\), where x is an element of a finite lattice. Based on these investigations we present an algorithmic procedure for the matrix computation of the small inductive dimension of an arbitrary finite lattice.

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Acknowledgements

The authors would like to thank the referees for the careful reading of the paper and the useful comments.

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Author notes

  1. R. B. Beshimov, D. Georgiou and F. Sereti contributed equally to this work.

Authors and Affiliations

  1. National University of Uzbekistan named after Mirzo Ulugbek, str. University, Tashkent, 100174, Uzbekistan

    R. B. Beshimov

  2. Department of Mathematics, University of Patras, Patra, 26500, Greece

    D. Georgiou

  3. Department of Mathematics, University of Western Macedonia, Kastoria, 52100, Greece

    F. Sereti

Authors
  1. R. B. Beshimov
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  2. D. Georgiou
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  3. F. Sereti
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Correspondence to D. Georgiou.

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Beshimov, R.B., Georgiou, D. & Sereti, F. The small inductive dimension of finite lattices through matrices. Comp. Appl. Math. 42, 145 (2023). https://doi.org/10.1007/s40314-023-02234-9

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  • DOI: https://doi.org/10.1007/s40314-023-02234-9

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