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

Table 1: Special categories of modules over a rig
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