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 Fundam. Prikl. Mat., 2008, Volume 14, Issue 5, Pages 3–54 (Mi fpm1138)

Semifields and their properties

E. M. Vechtomova, A. V. Cheranevab

a Vyatka State University of Humanities
b Vyatka State University

Abstract: An introduction to the theory of semifields is included in the first part of the article: basic concepts, initial properties, and several methods of investigating semifields are examined. Semifields with a generator, in particular bounded semifields, are considered. Elements of the theory of kernels of semifields are also included in the paper: the structure of principal kernels; the kernel generated by the element $2=1+1$; indecomposable and maximal spectra of semifields; properties of the lattice of kernels of a semifield. A fragment of arp-semiring theory, which is the basis of a new method in semifield theory, is also included in the first part. The second part of the work is devoted to sheaves of semifields and functional representations of semifields. Properties of semifields of sections of semifield sheaves over a zero-dimensional compact are described. Two structural sheaves of semifields, which are the analogs of Pierce and Lambek sheaves for rings, are constructed. These sheaves give isomorphic functional representations of arbitrary, strongly Gelfand, and biregular semifields. As a result, sheaf characterizations of strongly Gelfand, biregular, and Boolean semifields are obtained.

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English version:
Journal of Mathematical Sciences (New York), 2009, 163:6, 625–661

Bibliographic databases:

UDC: 512.55

Citation: E. M. Vechtomov, A. V. Cheraneva, “Semifields and their properties”, Fundam. Prikl. Mat., 14:5 (2008), 3–54; J. Math. Sci., 163:6 (2009), 625–661

Citation in format AMSBIB
\Bibitem{VecChe08} \by E.~M.~Vechtomov, A.~V.~Cheraneva \paper Semifields and their properties \jour Fundam. Prikl. Mat. \yr 2008 \vol 14 \issue 5 \pages 3--54 \mathnet{http://mi.mathnet.ru/fpm1138} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2533575} \elib{http://elibrary.ru/item.asp?id=12174983} \transl \jour J. Math. Sci. \yr 2009 \vol 163 \issue 6 \pages 625--661 \crossref{https://doi.org/10.1007/s10958-009-9717-3} \elib{http://elibrary.ru/item.asp?id=15311552} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-73249139449} 

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This publication is cited in the following articles:
1. V. V. Chermnykh, “Functional representations of semirings”, J. Math. Sci., 187:2 (2012), 187–267
2. E. M. Vechtomov, A. V. Mikhalev, V. V. Sidorov, “Semirings of continuous functions”, J. Math. Sci., 237:2 (2019), 191–244
3. Schwartz N., “Positive Semifields and Their Ideals”, Ordered Algebraic Structures and Related Topics, Contemporary Mathematics, 697, eds. Broglia F., Delon F., Dickmann M., GondardCozette D., Powers V., Amer Mathematical Soc, 2017, 301–323
4. Chermnykh V.V. Chermnykh O.V., “Functional Representations of Lattice-Ordered Semirings”, Sib. Electron. Math. Rep., 14 (2017), 946–971
5. Perri T. Rowen L.H., “Kernels in Tropical Geometry and a Jordan-Holder Theorem”, J. Algebra. Appl., 17:4 (2018), 1850066
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