Rigs

A rig (“ring without negatives”) is like a ring, except that additive inverses need not exist. To be precise, a rig is monoid in the monoidal category of commutative monoids. Rigs are important in their own right and also through their modules. A categorified rig is a rig category.

Examples

Examples of rigs which are not rings include:

  • Natural numbers N={0,1,… }\mathbb{N} = \{0,1,\dots\}
  • Nonnegative reals R+=[0,∞)\mathbb{R}_+ = [0,\infty)
  • Booleans B=({0,1},max,0,min,1)\mathbb{B} = (\{0,1\}, \mathrm{max}, 0, \mathrm{min}, 1)
  • Tropical rigs
  • Logarithmic rigs
  • Viterbi rigs
    • Goodman, 1999: Semiring parsing (pdf)

Literature

Rigs are officially known as semirings and much of the literature on them must be accessed through this cumbersome name.

Introductions and surveys

Books

  • Golan, 1992: The theory of semirings, with applications to mathematics and theoretical computer science
  • Golan, 1999: Semirings and their applications (doi)
    • Golan, 1999: Power algebras over semirings (doi)
    • Golan, 2003: Semirings and affine equations over them: Theory and applications (doi)
  • Głazek, 2002: A guide to the literature on semirings and their applications (doi)
    • Massive annotated bibliography, for the experts only
  • Gondran & Minoux, 2008: Graphs, dioids, and semirings: New models and applications (doi)