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Algebraic Properties of Parikh \(\texttt {q}\)-Matrices on Two-Dimensional Words

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Combinatorial Image Analysis (IWCIA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13348))

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Abstract

Based on the idea of \(\texttt {q}-\)count of certain subwords of a word, the notion of Parikh \(\texttt {q}-\)matrix of a word over an ordered alphabet was introduced. On the other hand, with a two-dimensional picture array of symbols arranged in rows and columns, two kinds of upper triangular matrices, known as row and column Parikh \(\texttt {q}-\)matrices have also been introduced and investigated. Certain algebraic properties such as Parikh \(\texttt {q}-\)matrices commutators, alternate Parikh \(\texttt {q}-\)matrices and extending Parikh \(\texttt {q}-\)matrices have been investigated for one dimensional case, yet they do not suffice for two-dimensional words. In this paper, we introduce Parikh \(\texttt {q}-\)matrices commutators, alternate Parikh \(\texttt {q}-\)matrices and extending Parikh \(\texttt {q}-\)matrices for two-dimensional words and discuss their properties. We also derive the result used for transferring information with respect to subword occurrences derived from Parikh \(\texttt {q}-\)matrices to corresponding information derived from extending Parikh \(\texttt {q}-\)matrices.

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Acknowledgements

We would like to thank the unknown referees for their comments and suggestions on the manuscript in improving from an earlier version. The authors K. Janaki and R. Arulprakasam are very much thankful to the management, SRM Institute of Science and Technology for their continuous support and encouragement.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Chennai, 603203, India

    K. Janaki & R. Arulprakasam

  2. FB IV - Informatikwissenschaften, Universität Trier, 54286, Trier, Germany

    Meenakshi Paramasivan

  3. Department of Mathematics, Madras Christian College, Chennai, 600059, India

    V. Rajkumar Dare

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  1. K. Janaki
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  2. R. Arulprakasam
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  3. Meenakshi Paramasivan
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  4. V. Rajkumar Dare
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Correspondence to R. Arulprakasam .

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Editors and Affiliations

  1. State University of New York at Fredonia, Fredonia, NY, USA

    Reneta P. Barneva

  2. SUNY Buffalo State College, Buffalo, NY, USA

    Valentin E. Brimkov

  3. University of Messina, Messina, Italy

    Giorgio Nordo

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Janaki, K., Arulprakasam, R., Paramasivan, M., Dare, V.R. (2023). Algebraic Properties of Parikh \(\texttt {q}\)-Matrices on Two-Dimensional Words. In: Barneva, R.P., Brimkov, V.E., Nordo, G. (eds) Combinatorial Image Analysis. IWCIA 2022. Lecture Notes in Computer Science, vol 13348. Springer, Cham. https://doi.org/10.1007/978-3-031-23612-9_11

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  • DOI: https://doi.org/10.1007/978-3-031-23612-9_11

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