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 \(\mathbb{N} = \{0,1,\dots\}\)
- Nonnegative reals \(\mathbb{R}_+ = [0,\infty)\)
- Booleans \(\mathbb{B} = (\{0,1\}, \mathrm{max}, 0, \mathrm{min}, 1)\)
- 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)
- GÅ‚azek, 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)