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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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
- Polynomial Function
- Commutative Semigroup