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    Analysis approach to finite monoids
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2013) Cevik, A. Sinan; Cangul, I. Naci; Simsek, Yilmaz
    In a previous paper by the authors, a new approach between algebra and analysis has been recently developed. In detail, it has been generally described how one can express some algebraic properties in terms of special generating functions. To continue the study of this approach, in here, we state and prove that the presentation which has the minimal number of generators of the split extension of two finite monogenic monoids has different sets of generating functions (such that the number of these functions is equal to the number of generators) that represent the exponent sums of the generating pictures of this presentation. This study can be thought of as a mixture of pure analysis, topology and geometry within the purposes of this journal. AMS Subject Classification: 11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.
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    A new approach to connect algebra with analysis: relationships and applications between presentations and generating functions
    (SPRINGEROPEN, 2013) Cangul, Ismail Naci; Cevik, Ahmet Sinan; Simsek, Yilmaz
    For a minimal group (or monoid) presentation P, let us suppose that P satisfies the algebraic property of either being efficient or inefficient. Then one can investigate whether some generating functions can be applied to it and study what kind of new properties can be obtained by considering special generating functions. To establish that, we will use the presentations of infinite group and monoid examples, namely the split extensions Z(n) X Zand Z(2) X Z, respectively. This study will give an opportunity to make a new classification of infinite groups and monoids by using generating functions.
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    Some array polynomials over special monoid presentations
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2013) Cevik, Ahmet Sinan; Das, Kinkar Chandra; Simsek, Yilmaz; Cangul, Ismail Naci
    In a recent joint paper (Cevik et al. in Hacet. J. Math. Stat., acceptted), the authors have investigated the p-Cockcroft property (or, equivalently, efficiency) for a presentation, say , of the semi-direct product of a free abelian monoid rank two by a finite cyclic monoid. Moreover, they have presented sufficient conditions on a special case for to be minimal whilst it is inefficient. In this paper, by considering these results, we first show that the presentations of the form can actually be represented by characteristic polynomials. After that, some connections between representative characteristic polynomials and generating functions in terms of array polynomials over the presentation will be pointed out. Through indicated connections, the existence of an equivalence among each generating function in itself is claimed studied in this paper. MSC: 11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.