On unitary analogs of GCD reciprocal LCM matrices |
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Authors: | Pentti Haukkanen Pauliina Ilmonen Ayse Nalli Juha Sillanpää |
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Institution: | 1. Department of Mathematics, Statistics and Philosophy , University of Tampere , Tampere 33014, Finland pentti.haukkanen@uta.fi;3. Department of Mathematics, Statistics and Philosophy , University of Tampere , Tampere 33014, Finland;4. Department of Mathematics , Selcuk University , Campus, Konya 42031, Turkey |
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Abstract: | A divisor d ∈ ?+ of n ∈ ?+ is said to be a unitary divisor of n if (d, n/d) = 1. In this article we examine the greatest common unitary divisor (GCUD) reciprocal least common unitary multiple (LCUM) matrices. At first we concentrate on the difficulty of the non-existence of the LCUM and we present three different ways to overcome this difficulty. After that we calculate the determinant of the three GCUD reciprocal LCUM matrices with respect to certain types of functions arising from the LCUM problematics. We also analyse these classes of functions, which may be referred to as unitary analogs of the class of semimultiplicative functions, and find their connections to rational arithmetical functions. Our study shows that it does make a difference how to extend the concept of LCUM. |
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Keywords: | GCD matrix LCM matrix unitary divisor meet semilattice semimultiplicative function rational arithmetical function |
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