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Polynomial functions over commutative semi-groups

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Let S be a commutative semigroup. We consider the semigroup P(S) with respect to composition of all transformations p: S → S of the form x → a,x → xn or x → axn (a S; n N) and the semigroup P(S) containing only elements of the last two forms. Since all polynomials over S have the form a, xn or ax these transformations are the so-called polynomial functions over S. We investigate the relationship between the structures of S and\(\bar P\)(S) resp. P(S) — a criterion on the commutativity of S has been shown by means of polynomial functions in 2.

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  1. [1]

    Clifford,A.H. and G.B.Preston,The algebraic theory of semigroups. Vol.I, Amer.Math.Soc., Providence 1961

  2. [2]

    Kowol,G.,Conditions for the commutativity of semi-groups, Proc.Amer.Math.Soc. — to appear

  3. [3]

    Lausch,H. and W. Nöbauer,Algebra of polynomials, North-Holland Publ.Comp., Amsterdam-London 1973

  4. [4]

    Lausch, H., W. Nöbauer und F. Schweiger,Polynompermutationen auf Grupren II, Monatsh. Math. 70 (1966), 118–126

  5. [5]

    Petrich, M.,Introduction to semigroups, Ch.E. Merill Publ.Comp., Columbus-Ohio 1973.

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Communicated by M. Petrich

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Kowol, G., Mitsch, H. Polynomial functions over commutative semi-groups. Semigroup Forum 12, 109–118 (1976). https://doi.org/10.1007/BF02195915

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  • Polynomial Function
  • Commutative Semigroup