Modules over rigs
A module over a rig \(R\), or \(R\)-module, is just like a module over a ring, but generalized to rigs. Because rigs are officially called semirings, modules over rigs are officially called semimodules .
Examples
Rig \(R\) | \(R\)-module |
---|---|
\(\mathbb{N}\) | Commutative monoid |
\(\mathbb{Z}\) | Abelian group |
\(\mathbb{R}_+\) | Conical space |
\(\mathbb{R}\) or \(\mathbb{C}\) | Real or complex vector space |
\(\mathbb{B} = \{0,1\}\) | Idempotent commutative monoid (bounded semilattice) |
Literature
Books and surveys
- Lawson, 2004: Idempotent analysis and continuous semilattices (doi, pdf)
- Survey paper on “idempotent analysis”: analysis with vector spaces replaced by modules over idempotent rigs
- Golan, 2003: Semirings and affine equations over them: Theory and applications, Ch. 7: Semimodules (doi)
- Baccelli et al, 1992: Synchronization and linearity: An algebra for discrete
event systems (pdf)
- Connections between tropical modules and graphs/Petri nets
Semimodules in category theory
- Heunen, 2008: Semimodule enrichment (doi, pdf)
- Heunen, 2009, PhD thesis: Categorical quantum models and logics (pdf), Sec. 2.5: Modules over rigs
- Willerton, 2013: Tight spans, Isbell completions and semi-tropical modules
(arxiv, pdf, nCat Cafe )
- Fuji, 2019: Enriched categories and tropical mathematics (arxiv)
Bilinear maps and inner products
- Heunen, 2009, PhD thesis, Section 3.5: Hilbert modules
- Tan, 2016: Inner products on semimodules (doi)
- Tan, 2014: Inner products on semimodules over a commutative semiring (doi)
- Wodzicki, 2012: Notes on measure and integration, and the underlying
structures (pdf)
- Paragraph 3.5.5.2 defines bilinear pairings for semimodules