Article
MSC:
08A05,
08A30,
08A72,
12K10,
13C60,
14T10,
16Y60,
18A05,
18C10,
18E05,
20N20 | MR 4538623 | Zbl 07655857 | DOI: 10.14736/kyb-2022-5-0733
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
pair; semiring; system; triple; shallow; algebraic; integral; affine; Ore; negation map; congruence; module
pair; semiring; system; triple; shallow; algebraic; integral; affine; Ore; negation map; congruence; module
Summary:
We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context.
References:
[1] Akian, M., Gaubert, S., Guterman, A.: Linear independence over tropical semirings and beyond. In: Tropical and Idempotent Mathematics (G. L. Litvinov and S. N. Sergeev, eds.), Contemp. Math. 495 (2009), 1-38. DOI 10.1090/conm/495/09689
[2] Akian, M., Gaubert, S., Rowen, L.: Linear algebra over systems. Preprint, 2022.
[3] Akian, M., Gaubert, S., Rowen, L.: From systems to hyperfields and related examples. Preprint, 2022.
[4] Alarcon, F., Anderson, D.: Commutative semirings and their lattices of ideals. J. Math. 20 (1994), 4.
[5] Baker, M., Bowler, N.: Matroids over partial hyperstructures. Adv. Math. 343 (2019), 821-863. DOI
[6] Bell, J., Zelmanov, E.: On the growth of algebras, semigroups, and hereditary languages. Inventiones Math. 224 (2021), 683-697. DOI
[7] Connes, A., Consani, C.: From monoids to hyperstructures: in search of an absolute arithmetic. Casimir Force, Casimir Operators and the Riemann Hypothesis, de Gruyter 2010, pp. 147-198. DOI
[8] Chapman, A., Gatto, L., Rowen, L.: Clifford semialgebras. Rendiconti del Circolo Matematico di Palermo Series 2, 2022 DOI
[9] Costa, A. A.: Sur la theorie generale des demi-anneaux. Publ. Math. Decebren 10 (1963), 14-29. DOI
[10] Dress, A.: Duality theory for finite and infinite matroids with coefficients. Advances Math. 93 (1986), 2, 214-250.
[11] Dress, A., Wenzel, W.: Algebraic, tropical, and fuzzy geometry. Beitrage zur Algebra und Geometrie/ Contributions to Algebra und Geometry 52 (2011), 2, 431-461. DOI
[12] Elizarov, N., Grigoriev, D.: A tropical version of Hilbert polynomial (in dimension one), (2021). DOI
[13] Gatto, L., Rowen, L.: Grassman semialgebras and the Cayley-Hamilton theorem. Proc. American Mathematical Society, series B, 7 (2020), 183-201. DOI
[14] Gaubert, S.: Theorie des systemes lineaires dans les diodes. These, Ecole des Mines de Paris 1992.
[15] Gaubert, S.: Methods and applications of (max,+) linear algebra. STACS' 97, number 1200 in LNCS, Lübeck, Springer 1997.
[16] Giansiracusa, J., Jun, J., Lorscheid, O.: On the relation between hyperrings and fuzzy rings. Beitr. Algebra Geom. 58 (2017), 735-764. DOI
[17] Golan, J.: The theory of semirings with applications in mathematics and theoretical computer science. Longman Sci Tech. 54 (1992).
[18] Greenfeld, B.: Growth of monomial algebras, simple rings and free subalgebras. J. Algebra 489 (2017), 427-434. DOI
[19] Hilgert, J., Hofmann, K.: Semigroups in Lie groups, semialgebras in Lie algebras. Trans. Amer. Math. Soc. 288 (1985), 2. DOI
[20] Izhakian, Z.: Tropical arithmetic and matrix algebra. Commun. Algebra 37 (2009), 4, 1445-1468. DOI
[21] Izhakian, Z., Rowen, L.: Supertropical algebra. Adv. Math. 225 (2010), 4, 2222-2286. DOI
[22] Izhakian, Z., Rowen, L.: Supertropical matrix algebra. Israel J. Math. 182 (2011), 1, 383-424. DOI
[23] Jacobson, N.: Basic Algebra II. Freeman 1980.
[24] Joo, D., Mincheva, K.: Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials. Selecta Mathematica 24 (2018), 3, 2207-2233. DOI
[25] Jun, J., Mincheva, K., Rowen, L.: Projective systemic modules. J. Pure Appl. Algebra 224 (2020), 5, 106-243.
[26] Jun, J.: Algebraic geometry over hyperrings. Adv. Math. 323 (2018), 142-192. DOI
[27] Jun, J., Rowen, L.: Categories with negation. In: Categorical, Homological and Combinatorial Methods in Algebra (AMS Special Session in honor of S. K. Jain's 80th birthday), Contempor. Math. 751 (2020), 221-270.
[28] Katsov, Y.: Tensor products of functors. Siberian J. Math. 19 (1978), 222-229, trans. from Sib. Mat. Zhurnal 19 (1978), 2, 318-327. DOI 10.1007/BF00970503
[29] Krasner, M.: A class of hyperrings and hyperfields. Int. J. Math. Math. Sci. 6 (1983), 2, 307-312. DOI 10.1155/S0161171283000265
[30] Krause, G. R., Lenagan, T. H.: Growth of algebras and Gelfand-Kirillov dimension. Amer. Math. Soc. Graduate Stud. Math. 22 (2000).
[31] Ljapin, E. S.: Semigroups. AMS Translations of Mathematical Monographs 3 (1963), 519 pp.
[32] Lorscheid, O.: The geometry of blueprints. Part I. Adv. Math. 229 (2012), 3, 1804-1846. DOI
[33] Lorscheid, O.: A blueprinted view on $\mathbb{F}_1$-geometry. In: Absolute Arithmetic and $\mathbb{F}_1$-geometry (Koen Thas. ed.), European Mathematical Society Publishing House 2016.
[34] Rowen, L. H.: Ring Theory. Vol. I. Academic Press, Pure and Applied Mathematics 127, 1988.
[35] Rowen, L. H.: Graduate algebra: Noncommutative View. AMS Graduate Studies in Mathematics 91, 2008.
[36] Rowen, L. H.: Algebras with a negation map. Europ. J. Math. 8 (2022), 62-138. DOI 10.1007/s40879-021-00499-0
[37] Rowen, L. H.: An informal overview of triples and systems, 2017. DOI
[38] Smoktunowicz, A.: Growth, entropy and commutativity of algebras satisfying prescribed relations. Selecta Mathematica 20 (2014) 4, 1197-1212. DOI
[39] Shneerson, L. M.: Types of growth and identities of semigroups. Int. J. Algebra Comput., Special Issue: International Conference on Group Theory "Combinatorial, Geometric and Dynamical Aspects of Infinite Groups"; (L. Bartholdi, T. Ceccherini-Silberstein, T. Smirnova-Nagnibeda, A. Zuk, eds.), 15 (2005), 05, 1189-1204. DOI
[40] Viro, O. Y.: Hyperfields for tropical geometry I, Hyperfields and dequantization, 2010. DOI
Search
Partner of
