Semifields and their properties |
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Authors: | E M Vechtomov A V Cheraneva |
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Institution: | 1.Vyatka State University of Humanities,Vyatka,Russia;2.Vyatka State University,Vyatka,Russia |
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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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