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Lattice-Ordered Semirings

In: Semirings and their Applications

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  • Jonathan S. Golan

    (University of Haifa)

Abstract

A semiring R is lattice-ordered if and only if it also has the structure of a lattice such that, for all a and b in R: (1) a + b = a ∨ b; and (2) ab ≤ a ∧ b, where partial order here is the one induced naturally by the lattice structure on R. If R, as a lattice, is distributive, then R is a distributive lattice-ordered semiring (DLO-semiring). Clearly any lattice-ordered semiring is a partially-ordered semiring in the sense of Chapter 18, with respect to the partial order induced by the lattice structure. (Note in passing that some authors replace (1) by a weaker condition; see, for example, [Ranga Rao, 1981].)

Suggested Citation

  • Jonathan S. Golan, 1999. "Lattice-Ordered Semirings," Springer Books, in: Semirings and their Applications, chapter 21, pages 239-246, Springer.
    • Handle: RePEc:spr:sprchp:978-94-015-9333-5_21
    DOI: 10.1007/978-94-015-9333-5_21
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