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Polynomial functions over commutative semi-groups

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Abstract

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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References

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    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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Keywords

  • Polynomial Function
  • Commutative Semigroup