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Ordered Rings of Generalized Power Series

In: Ordered Algebraic Structures

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  • A. Benhissi
  • P. Ribenboim

Abstract

In this paper, we consider orders on rings of generalized power series. Unless the contrary is expressly stated, we do not assume the orders to be total (=linear); for brevity we omit the qualification “partial” order. The first section deals with the order introduced by Conrad, Harvey & Holland on abelian additive groups of maps from an ordered set (S, ≤) to an ordered abelian group (R, ≤), having artinian support. In particular, we note the conditions to obtain a lattice ordered group. In §2, it is proved that if (S ≤) is a strictly ordered monoid and (R, ≤) a strictly ordered ring, the order defined on the ring of generalized power series A = [[R s,≤ ]]is compatible and strict. Finally, in §3 conditions are indicated for an ordered monoid to be a tree. Then, under appropriate conditions, A is a lattice-ordered ring. To conclude the paper, many examples of lattice ordered rings of generalized power series are provided.

Suggested Citation

  • A. Benhissi & P. Ribenboim, 1993. "Ordered Rings of Generalized Power Series," Springer Books, in: J. Martinez & C. Holland (ed.), Ordered Algebraic Structures, pages 99-109, Springer.
    • Handle: RePEc:spr:sprchp:978-94-011-1723-4_7
    DOI: 10.1007/978-94-011-1723-4_7
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