×

\(\mathcal{T}\)-semiring pairs. (English) Zbl 07655857

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.

MSC:

08A05 Structure theory of algebraic structures
14T10 Foundations of tropical geometry and relations with algebra
16Y60 Semirings
18A05 Definitions and generalizations in theory of categories
18C10 Theories (e.g., algebraic theories), structure, and semantics
08A30 Subalgebras, congruence relations
08A72 Fuzzy algebraic structures
12K10 Semifields
13C60 Module categories and commutative rings
18E05 Preadditive, additive categories
20N20 Hypergroups
PDFBibTeX XMLCite
Full Text: DOI arXiv

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 · Zbl 1182.15002
[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 · Zbl 0823.16032
[5] Baker, M.; Bowler, N., Matroids over partial hyperstructures., Adv. Math. 343 (2019), 821-863
[6] Bell, J.; Zelmanov, E., On the growth of algebras, semigroups, and hereditary languages., Inventiones Math. 224 (2021), 683-697
[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 · Zbl 1234.14002
[8] Chapman, A.; Gatto, L.; Rowen, L., Clifford semialgebras., Rendiconti del Circolo Matematico di Palermo Series 2, 2022
[9] Costa, A. A., Sur la theorie generale des demi-anneaux., Publ. Math. Decebren 10 (1963), 14-29 · Zbl 0135.02802
[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 · Zbl 1246.14079
[12] Elizarov, N.; Grigoriev, D., A tropical version of Hilbert polynomial (in dimension one), (2021).
[13] Gatto, L.; Rowen, L., Grassman semialgebras and the Cayley-Hamilton theorem., Proc. American Mathematical Society, series B, 7 (2020), 183-201
[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 · Zbl 1498.15034
[16] Giansiracusa, J.; Jun, J.; Lorscheid, O., On the relation between hyperrings and fuzzy rings., Beitr. Algebra Geom. 58 (2017), 735-764 · Zbl 1388.14018
[17] Golan, J., The theory of semirings with applications in mathematics and theoretical computer science., Longman Sci Tech. 54 (1992) · Zbl 0780.16036
[18] Greenfeld, B., Growth of monomial algebras, simple rings and free subalgebras., J. Algebra 489 (2017), 427-434 · Zbl 1380.16020
[19] Hilgert, J.; Hofmann, K., Semigroups in Lie groups, semialgebras in Lie algebras., Trans. Amer. Math. Soc. 288 (1985), 2 · Zbl 0565.22007
[20] Izhakian, Z., Tropical arithmetic and matrix algebra., Commun. Algebra 37 (2009), 4, 1445-1468 · Zbl 1165.15017
[21] Izhakian, Z.; Rowen, L., Supertropical algebra., Adv. Math. 225 (2010), 4, 2222-2286 · Zbl 1273.14132
[22] Izhakian, Z.; Rowen, L., Supertropical matrix algebra., Israel J. Math. 182 (2011), 1, 383-424 · Zbl 1215.15018
[23] Jacobson, N., Basic Algebra II., Freeman 1980 · Zbl 0441.16001
[24] Joo, D.; Mincheva, K., Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials., Selecta Mathematica 24 (2018), 3, 2207-2233
[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 · Zbl 1420.14005
[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 · Zbl 1462.08001
[28] Katsov, Y., Tensor products of functors., Siberian J. Math. 19 (1978), 222-229, trans. from Sib. Mat. Zhurnal 19 (1978), 2, 318-327
[29] Krasner, M., A class of hyperrings and hyperfields., Int. J. Math. Math. Sci. 6 (1983), 2, 307-312 · Zbl 0516.16030
[30] Krause, G. R.; Lenagan, T. H., Growth of algebras and Gelfand-Kirillov dimension., Amer. Math. Soc. Graduate Stud. Math. 22 (2000) · Zbl 0957.16001
[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 · Zbl 1259.14001
[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 · Zbl 1351.11041
[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 · Zbl 1182.16001
[36] Rowen, L. H., Algebras with a negation map., Europ. J. Math. 8 (2022), 62-138 · Zbl 1516.14112
[37] Rowen, L. H., An informal overview of triples and systems, 2017.
[38] Smoktunowicz, A., Growth, entropy and commutativity of algebras satisfying prescribed relations., Selecta Mathematica 20 (2014) 4, 1197-1212 · Zbl 1320.16009
[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 · Zbl 1104.20052
[40] Viro, O. Y., Hyperfields for tropical geometry I, Hyperfields and dequantization, 2010.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.