Categories of Layered Semirings

REFERENCES

• Akian , M. , Bapat , R. , Gaubert , S. ( 2006 ). Max-plus algebra . In: Hogben , L. , Brualdi , R. , Greenbaum , A. , Mathias , R. eds. Handbook of Linear Algebra . London : Chapman and Hall .
   Google Scholar
• Akian , M. , Gaubert , S. , Guterman , A. (2009). Linear independence over tropical semirings and beyond. In: Litvinov , G. L. , Sergeev , S. N. eds. Tropical and Idempotent Mathematics, Contemp. Math. 495:1–38.
   Google Scholar
• Berkovich , V. ( 1990 ). Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Mathematical Surveys and Monographs. Vol. 33. Amer. Math. Soc .
   Google Scholar
• Connes , A. , Consani , C. ( 2009 ). Characteristic 1, entropy, and the absolute point. Preprint at arXiv:math.0911.3537 .
   Google Scholar
• Costa , A. A. ( 1963 ). Sur la thêorie générale des demi-anneaux . Publ. Math. Decebren 10 : 14 – 29 .
   Google Scholar
• Cortinas , G. , Haesemeyer , C. , Walker , M. , Weibel , C. ( 2010 ). Toric varieties, monoid schemes, and descent (preprint) .
   Google Scholar
• Dickenstein , A. , Feichtner , E. M. , Sturmfels , B. ( 2007 ). Tropical discriminants . J. Amer. Math. Soc ( 20 ): 1111 – 1133 .
   Web of Science ®Google Scholar
• Dubey , M. Some results on semimodules analogous to module theory. Doctoral dissertation. University of Delhi, India.
   Google Scholar
• Golan , J. ( 1992 ). The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science. Vol 54. Longman Sci. & Tech .
   Google Scholar
• Harrison , D. K. , Vitulli , M. A. ( 1989 ). V-valuations of a commutative ring I . J. Algebra 126 : 264 – 292 .
   Web of Science ®Google Scholar
• Hartshorne , R. ( 1977 ). Algebraic Geometry. Number 52 in Graduate Texts in Mathematics, Springer .
   Google Scholar
• Hutchins , H. C. , Weinert , H. J. ( 1990 ). Homomorphisms and kernels of semifields . Periodica Mathematica Hungaria 21 ( 2 ): 113 – 152 .
   Google Scholar
• Itenberg , I. , Mikhalkin , G. , Shustin , E. ( 2007 ). Tropical Algebraic Geometry. Oberwolfach Seminars. Vol. 35. Birkhauser .
   Google Scholar
• Izhakian , Z. ( 2009 ). Tropical arithmetic and matrix algebra . Comm. in Algebra 37 ( 4 ): 1445 – 1468 .
   Web of Science ®Google Scholar
• Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Appl. Alg. 215 ( 10 ): 2431 – 2463 .
   Web of Science ®Google Scholar
• Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2014 ). Layered tropical mathematics . J. Algebra 416 : 200 – 273 .
   Web of Science ®Google Scholar
• Izhakian , Z. , Knebusch , M. , Rowen , L. Dominance and transmissions in supertropical valuation theory . Comm. in Algebra to appear (Preprint at arXiv:1102.1520.)
   Web of Science ®Google Scholar
• Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2013 ). Categorical notions of layered tropical algebra and geometry . In: Brugalle , E. , Cueto , M. A. , Dickenstein , A. , Feichtner , E.-M. , Itenberg , I. , eds. Algebraic and Combinatorial Aspects of Tropical Geometry . Contemporary Mathematics on Tropical Geometry, AMS, Vol. 589, pp. 191–234 .
   Google Scholar
• Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical monoids: Basics, canonical factorization, and lifting ghosts to tangibles. preprint at arXiv:1108.1880 .
   Google Scholar
• Izhakian , Z. , Rowen , L. ( 2010 ). Supertropical algebra . Adv. in Math. 324 ( 8 ): 1860 – 1886 .
   Google Scholar
• Izhakian , Z. , Rowen , L. ( 2011 ). Supertropical matrix algebra . Israel J. Math. 182 ( 1 ): 383 – 424 .
   Web of Science ®Google Scholar
• Jacobson , N. ( 1980 ). Basic Algebra II. Freeman .
   Google Scholar
• Litvinov , G. ( 2007 ). The Maslov dequantization, idempotent and tropical mathematics: A very brief introduction . J. Math. Sciences 140 ( 3 ): 426 – 444 .
   Google Scholar
• Maclagan , D. , Sturmfels , B. ( 2015 ). Introduction to Tropical Geometry . Graduate Studies in Mathematics , Vol. 161 . Providence , RI : American Mathematical Society .
   Google Scholar
• Patchkoria , A. ( 1986 ). On derived functors of semimodule-valued functors . Proc. A. Razmadze Math. Inst. 83 : 60 – 75 (in Russian) .
   Google Scholar
• Payne , S. ( 2009 ). Fibers of tropicalization . Math Z. 262 ( 2 ): 301 – 311 .
   Web of Science ®Google Scholar
• Payne , S. ( 2009 ). Analytification is the limit of all tropicalizations . Math. Res. Lett. 16 ( 3 ): 543 – 556 .
   Web of Science ®Google Scholar
• Rowen , L. (2006). Graduate Algebra: Commutative View. Graduate Studies in Mathematics. Vols. 73, 91. American Mathematical Society.
   Google Scholar
• Rowen , L. ( 2006 ). Graduate Algebra: Noncommutative View. Graduate Studies in Mathematics. Vol. 73. American Mathematical Society .
   Google Scholar
• Zhang , D. ( 2002 ). The M-valuation spectrum of a commutative ring . Comm. in Algebra 30 : 2883 – 2896 .
   Web of Science ®Google Scholar
• Communicated by M. Cohen.
   Google Scholar