Home Positive rigs
Article
Licensed
Unlicensed Requires Authentication

Positive rigs

  • Matías Menni EMAIL logo
Published/Copyright: November 30, 2023
Forum Mathematicum
From the journal Forum Mathematicum Volume 36 Issue 4

Abstract

A positive rig is a commutative and unitary semi-ring A such that 1 + x is invertible for every x ∈ A . We show that the category of positive rigs shares many properties with that of K-algebras for a (non-algebraically closed) field K. In particular, it is coextensive and, although we do not have an analogue of Hilbert’s basis theorem for positive rigs, we show that every finitely presentable positive rig is a finite direct product of directly indecomposable ones. We also describe free positive rigs as rigs of rational functions with non-negative rational coefficients, and we give a characterization of the positive rigs with a unique maximal ideal.

Keywords: (Co)extensive categories; rigs; positivity; commutative algebra
MSC 2020: 16Y60; 06F25; 13J25

Communicated by Manfred Droste

Funding source: H2020 Marie Skłodowska-Curie Actions

Award Identifier / Grant number: 101007627

Funding source: Consejo Nacional de Investigaciones Científicas y Técnicas

Award Identifier / Grant number: PIP 11220200100912CO

Funding statement: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101007627, and also from CONICET (Argentina), PIP 11220200100912CO.

References

[1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, 1969. Search in Google Scholar

[2] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergeb. Math. Grenzgeb. (3) 36, Springer, Berlin, 1998. 10.1007/978-3-662-03718-8Search in Google Scholar

[3] N. Bourbaki, Algebra II. Chapters 4–7, Elem. Math. (Berlin), Springer, Berlin, 2003. 10.1007/978-3-642-61698-3Search in Google Scholar

[4] S. Bourne, On the radical of a positive semiring, Proc. Natl. Acad. Sci. USA 45 (1959), 10.1073/pnas.45.10.1519. 10.1073/pnas.45.10.1519Search in Google Scholar PubMed PubMed Central

[5] A. Carboni, S. Lack and R. F. C. Walters, Introduction to extensive and distributive categories, J. Pure Appl. Algebra 84 (1993), no. 2, 145–158. 10.1016/0022-4049(93)90035-RSearch in Google Scholar

[6] A. Carboni, M. C. Pedicchio and J. Rosický, Syntactic characterizations of various classes of locally presentable categories, J. Pure Appl. Algebra 161 (2001), no. 1–2, 65–90. 10.1016/S0022-4049(01)00016-0Search in Google Scholar

[7] J. L. Castiglioni, M. Menni and W. J. Zuluaga Botero, A representation theorem for integral rigs and its applications to residuated lattices, J. Pure Appl. Algebra 220 (2016), no. 10, 3533–3566. 10.1016/j.jpaa.2016.04.014Search in Google Scholar

[8] V. V. Chermnykh, Representation of positive semirings by sections, Uspekhi Mat. Nauk 47 (1992), no. 5(287), 193–194. 10.1070/RM1992v047n05ABEH000948Search in Google Scholar

[9] M. Droste and W. Kuich, Chapter 1: Semirings and formal power series, Handbook of Weighted Automata, edited by M. Droste, W. Kuich and H. Vogler, Monogr. Theoret. Comput. Sci. EATCS Ser., Springer, Berlin (2009), 3–28. 10.1007/978-3-642-01492-5_1Search in Google Scholar

[10] J. S. Golan, Semirings and Their Applications, Kluwer Academic, Dordrecht, 1999. 10.1007/978-94-015-9333-5Search in Google Scholar

[11] P. T. Johnstone, Stone Spaces, Cambridge Stud. Adv. Math. 3, Cambridge University, Cambridge, 1982. Search in Google Scholar

[12] P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium. Vol. 1, Oxford Logic Guides 43, Oxford University, New York, 2002. 10.1093/oso/9780198515982.003.0004Search in Google Scholar

[13] A. Kock, Synthetic Differential Geometry, 2nd ed., London Math. Soc. Lecture Note Ser. 333, Cambridge University, Cambridge, 2006. Search in Google Scholar

[14] J.-L. Krivine, Anneaux préordonnés, J. Anal. Math. 12 (1964), 307–326. 10.1007/BF02807438Search in Google Scholar

[15] F. W. Lawvere, Grothendieck’s 1973 Buffalo Colloquium, 2003. Email to the categories list, March 2003. Search in Google Scholar

[16] F. W. Lawvere, Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories, Repr. Theory Appl. Categ. (2004), no. 5, 1–121. Search in Google Scholar

[17] S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic, Universitext, Springer, New York, 1992. Search in Google Scholar

[18] P. Mayr and N. Ruškuc, Finiteness properties of direct products of algebraic structures, J. Algebra 494 (2018), 167–187. 10.1016/j.jalgebra.2017.09.035Search in Google Scholar

[19] M. Menni, A basis theorem for 2-rigs and rig geometry, Cah. Topol. Géom. Différ. Catég. 62 (2021), no. 4, 451–490. Search in Google Scholar

[20] S. H. Schanuel, Negative sets have Euler characteristic and dimension, Category Theory (Como 1990), Lecture Notes in Math. 1488, Springer, Berlin (1991), 379–385. 10.1007/BFb0084232Search in Google Scholar

[21] W. Słowikowski and W. Zawadowski, A generalization of maximal ideals method of Stone and Gelfand, Fund. Math. 42 (1955), 215–231. 10.4064/fm-42-2-215-231Search in Google Scholar

[22] F. A. Smith, A structure theory for a class of lattice ordered semirings, Fund. Math. 59 (1966), 49–64. 10.4064/fm-59-1-49-64Search in Google Scholar

[23] W. J. Zuluaga Botero, Coextensive varieties via central elements, Algebra Universalis 82 (2021), no. 3, Paper No. 50. 10.1007/s00012-021-00745-2Search in Google Scholar
Received: 2022-09-18
Revised: 2023-08-15
Published Online: 2023-11-30
Published in Print: 2024-07-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The C*-algebra of the Boidol group
  3. Profinite genus of fundamental groups of compact flat manifolds with the cyclic holonomy group of square-free order
  4. Positive rigs
  5. Torus bundles over lens spaces
  6. Topological amenability of semihypergroups
  7. On projections of the tails of a power
  8. Li–Yorke chaos for composition operators on Orlicz spaces
  9. A note on the post quantum-Sheffer polynomial sequences
  10. Finite rigid sets of the non-separating curve complex
  11. Building planar polygon spaces from the projective braid arrangement
  12. Octonionic monogenic and slice monogenic Hardy and Bergman spaces
  13. Transcendence on algebraic groups
  14. An explicit version of Bombieri’s log-free density estimate and Sárközy’s theorem for shifted primes
  15. The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras
  16. Joint distribution of the cokernels of random p-adic matrices II
https://doi.org/10.1515/forum-2022-0271
Keywords for this article
(Co)extensive categories; rigs; positivity; commutative algebra

Articles in the same Issue

  1. Frontmatter
  2. The C*-algebra of the Boidol group
  3. Profinite genus of fundamental groups of compact flat manifolds with the cyclic holonomy group of square-free order
  4. Positive rigs
  5. Torus bundles over lens spaces
  6. Topological amenability of semihypergroups
  7. On projections of the tails of a power
  8. Li–Yorke chaos for composition operators on Orlicz spaces
  9. A note on the post quantum-Sheffer polynomial sequences
  10. Finite rigid sets of the non-separating curve complex
  11. Building planar polygon spaces from the projective braid arrangement
  12. Octonionic monogenic and slice monogenic Hardy and Bergman spaces
  13. Transcendence on algebraic groups
  14. An explicit version of Bombieri’s log-free density estimate and Sárközy’s theorem for shifted primes
  15. The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras
  16. Joint distribution of the cokernels of random p-adic matrices II
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 22.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2022-0271/html?lang=en
Scroll to top button