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In [G. Lunardon, Semifields and linear sets of PG(1,qt), Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta (in press)], G. Lunardon has exhibited a construction method yielding a theoretical family of semifields of order q2n,n>1 and n odd, with left nucleus Fqn, middle and right nuclei both Fq2 and center Fq. When n=3 this method gives an alternative construction of a family of semifields described in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti, On a generalization of cyclic semifields, J. Algebraic Combin. 26 (2009), 1-34], which generalizes the family of cyclic semifields obtained by Jha and Johnson in [V. Jha, N.L. Johnson, Translation planes of large dimension admitting non-solvable groups, J. Geom. 45 (1992), 87-104]. For n>3, no example of a semifield belonging to this family is known.In this paper we first prove that, when n>3, any semifield belonging to the family introduced in the second work cited above is not isotopic to any semifield of the family constructed in the former. Then we construct, with the aid of a computer, a semifield of order 210 belonging to the family introduced by Lunardon, which turns out to be non-isotopic to any other known semifield. 相似文献
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Rita Vincenti 《Journal of Geometry》1988,32(1-2):169-191
Derived semifield planes of odd order admitting non-trivial involutorial affine homologies with more than one axis, are examined in detail, under the assumption that the group generated in the translation complement is dihedral. The whole structure of the semifields S coordinatizing such planes is determined. The class of the semifields S of dimension 4 over their centres is characterized.Dedicated to A. Barlotti on the occasion of his 65. birthday.Research partially supported by G.N.S.A.G.A. (C.N.R.) 相似文献
4.
In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array (aijk) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions (f,g) where f and g are functions from GF(q) to GF(q) with the property that f and g are linear over some subfield and g(x)2+4xf(x) is a non-square for all x∈GF(q)∗, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4,q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f,g) are of linear type or of Dickson-Kantor-Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations. 相似文献
5.
Similarity of certain abelian collineation groups of a translation plane corresponds to strong isotopy of multiplication variations of a commutative semifield. Strong isotopy of Dickson semifields and their multiplication variations is characterized. The splitting of the isotopy class of a Dickson semifield over different types of basic fields (e. g. absolutely algebraic fields, number fields) into classes of strong isotopy is investigated. ** 相似文献
6.
Robert S. Coulter Marie Henderson Pamela Kosick 《Designs, Codes and Cryptography》2007,44(1-3):275-286
We consider the implications of the equivalence of commutative semifields of odd order and planar Dembowski-Ostrom polynomials. This equivalence was outlined recently by Coulter and Henderson. In particular, following a more general statement concerning semifields we identify a form of planar Dembowski-Ostrom polynomial which must define a commutative semifield with the nuclei specified. Since any strong isotopy class of commutative semifields must contain at least one example of a commutative semifield described by such a planar polynomial, to classify commutative semifields it is enough to classify planar Dembowski-Ostrom polynomials of this form and determine when they describe non-isotopic commutative semifields. We prove several results along these lines. We end by introducing a new commutative semifield of order 38 with left nucleus of order 3 and middle nucleus of order 32. 相似文献
7.
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. 相似文献
8.
In 1960 Hughes and Kleinfeld (Am J Math 82:389–392, 1960) constructed a finite semifield which is two-dimensional over a weak
nucleus, given an automorphism σ of a finite field and elements with the property that has no roots in . In 1965 Knuth (J Algebra 2:182–217, 1965) constructed a further three finite semifields which are also two-dimensional over
a weak nucleus, given the same parameter set . Moreover, in the same article, Knuth describes operations that allow one to obtain up to six semifields from a given semifield.
We show how these operations in fact relate these four finite semifields, for a fixed parameter set, and yield at most five
non-isotopic semifields out of a possible 24. These five semifields form two sets of semifields, one of which consists of
at most two non-isotopic semifields related by Knuth operations and the other of which consists of at most three non-isotopic
semifields.
相似文献
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Using ‘fusion’ methods on finite semifields, a variety of partitions (flocks) of Segre varieties by caps are obtained. The
partitions arise from semifield planes and are thus called “semifield flat flocks”. Furthermore, the finite transitive semifield
flat flocks are completely determined. 相似文献
11.
Ulrich Dempwolff 《Designs, Codes and Cryptography》2013,68(1-3):81-103
In this article we use pairs of Dembowski–Ostrom polynomials with special properties (see (P1)–(P3) in the introduction below) to construct translation planes of order q n which admit cyclic groups of order q n ?1 having orbits of lengths 1, 1, (q n ?1)/2, (q n ?1)/2 on the line at infinity. The same pairs also define semifields of order q 2n . We discuss the properties of these translation planes and semifields. These constructions extend the related construction in Dempwolff and Müller, Osaka J Math [5]. 相似文献
12.
Fractional dimensions in semifields of odd order 总被引:1,自引:0,他引:1
A finite semifield D is considered a fractional dimensional semifield if it contains a subsemifield E such that λ := log|E||D| is not an integer. We develop spread-theoretic tools to determine when finite planes admit coordinatization by fractional
semifields, and to find such semifields when they exist. We use our results to show that such semifields exist for prime powers
3
n
whenever n is an odd integer divisible by 5 or 7. 相似文献
13.
We construct and describe the basic properties of a family of semifields in characteristic 2. The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nuclei. The non-degenerate members of our family display a very different behavior. Their left and right nuclei agree with the center, the middle nucleus is quadratic over the center. None of those semifields is isotopic or Knuth equivalent to a commutative semifield. As a by-product we obtain the complete taxonomy of the characteristic 2 semifields which are quadratic over the middle nucleus, bi-quadratic over the left and right nuclei and not isotopic to twisted fields. This includes determining when two such semifields are isotopic and the order of the autotopism group. 相似文献
14.
In 1960, Kleinfeld published representatives for all of the isomorphism classes of 16 element semifields, [E. Kleinfeld, Techniques for Enumerating Veblen-Wedderburn Systems, J. ACM 7 (1960) 330–337]. It is not entirely clear how Kleinfeld generated some of his results, but it is likely that it was similar to the approach that Walker used in 1962 to generate representative for the isotopism classes of 32 element semifields, [R. J. Walker, Determination of Division Algebras With 32 Elements, Proc. Sympos. Appl. Math. XV (1963) 8385]. This paper introduces an alternative notation for publication which is both simple and practical, and which leads to an alternative method which was used to verify Kleinfeldʼs results. 相似文献
15.
In this article we introduce the notion of the BEL-rank of a finite semifield, prove that it is an invariant for the isotopism classes, and give geometric and algebraic interpretations of this new invariant. Moreover, we describe an efficient method for calculating the BEL-rank, and present computational results for all known small semifields. 相似文献
16.
Norman L. Johnson Giuseppe Marino Olga Polverino Rocco Trombetti 《Journal of Algebraic Combinatorics》2009,29(1):1-34
A new construction is given of cyclic semifields of orders q
2n
, n odd, with kernel (left nucleus)
and right and middle nuclei isomorphic to
, and the isotopism classes are determined. Furthermore, this construction is generalized to produce potentially new semifields
of the same general type that are not isotopic to cyclic semifields. In particular, a new semifield plane of order 45 and new semifield planes of order 165 are constructed by this method. 相似文献
17.
Albert's construction for commutative semifields of order 2 n , n odd, is presented. It avoids the construction of a presemifield and, in the case that n is prime, allows us to determine automorphism groups and the isomorphism classes. If n is a prime greater than three, the semifields are strictly not associative. These semifields are new for all n greater than three, differing from the binary semifields in that each admits only the trivial automorphism. The authors present an explicit construction of an isotope of the 25-element semifield that contains a subsemifield of order 22. 相似文献
18.
The lattices of congruences of semirings and semifields of continuous nonnegative functions over an arbitrary topological space are studied. It is proved that congruences of the semifield of continuous positive functions can be extended to congruences of the semiring of continuous nonnegative functions. 相似文献
19.
Guglielmo Lunardon 《Journal of Geometry》2003,76(1-2):200-215
Translation ovoids of have been recently studied by many
authors because of their connection with semifield flocks and
translation generalized quadrangles. In this paper we discuss some
recent results using the relationship between semifield spreads of and translation ovoids of as a guideline for our
approach. 相似文献