FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2008, VOLUME 14, NUMBER 5, PAGES 3-54

**Semifields and their properties**

E. M. Vechtomov

A. V. Cheraneva

Abstract

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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.

Location: http://mech.math.msu.su/~fpm/eng/k08/k085/k08501h.htm

Last modified: May 6, 2009