Finite Semifield Research Articles

On some 3-primitive projective planes

We evolve an approach to construction and classification of semifield projective planes with the use of the linear space and spread set. This approach is applied to the problem of existance for a projective plane with the fixed restrictions on collineation group. A projective plane is said to be semifield plane if its coordinatizing set is a semifield, or division ring. It is an algebraic structure with two binary operation which satisfies all the axioms for a skewfield except (possibly) associativity of multiplication. A collineation of a projective plane of order p 2n ( p > 2 be prime) is called Baer collineation if it fixes a subplane of order p n pointwise. If the order of a Baer collineation divides p n − 1 but does not divide p i − 1 for i < n then such a collineation is called p -primitive. A semifield plane that admit such collineation is a p -primitive plane. M. Cordero in 1997 construct 4 examples of 3-primitive semifield planes of order 81 with the nucleus of order 9, using a spread set formed by 2 × 2-matrices. In the paper we consider the general case of 3-primitive semifield plane of order 81 with the nucleus of order ≤ 9 and a spread set in the ring of 4 × 4-matrices. We use the earlier theoretical results obtained independently to construct the matrix representation of the spread set and autotopism group. We determine 8 isomorphism classes of 3-primitive semifield planes of order 81 including M. Cordero examples. We obtain the algorithm to optimize the identification of pair-isomorphic semifield planes, and computer realization of this algorithm. It is proved that full collineation group of any semifield plane of order 81 is solvable, the orders of all autotopisms are calculated. We describe the structure of 8 non-isotopic semifields of order 81 that coordinatize 8 nonisomorphic 3-primitive semifield planes of order 81. The spectra of its multiplicative loops of non-zero elements are calculated, the left-, right-ordered spectra, the maximal subfields and automorphisms are found. The results obtained illustrate G. Wene hypothesis on left or right primitivity for any finite semifield and demonstrate some anomalous properties. The methods and algorithsm demonstrated can be used for construction and investigation of semifield planes of odd order p n for p ≥ 3 and n ≥ 4.

The BEL-rank of finite semifields

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.

On the primitivity of four-dimensional finite semifields

We prove that any finite semifield of dimension four over its center Fq is right and left primitive, provided that q is odd and sufficiently large.

Permutations of cubical arrays

The structure constants of an algebra determine a cube called the cubical array associated with the algebra. The permuted indices of the cubical array associated with a finite semifield generate new division algebras. We do not not require that the algebra be finite and ask "Is it possible to choose a basis for the algebra such any permutation of the indices of the structure constants leaves the algebra unchanged?" What are the associated algebras? Author shows that the property "weakly quadratic" is invariant under all permutations of the indices of the corresponding cubical array and presents two algebras for which the cubical array is invariant under all permutations of the indices.

New examples of fractional dimensional semifield planes

Abstract This is a short round up of the fractional dimension of finite semifields and finite semifield planes. We show that the classical Knuth binary semifields can never be fractional dimensional and a special class of the generalized Knuth binary semifields is fractional dimensional, even though these two classes of semifields are isotopic. Finally we show that the Albert semifield A n ( S ) , where n is any odd integer, does not contain G F ( 2 3 ) when ( n , 3 ) = 1 .

Cyclic semirings with idempotent noncommutative addition

The article discusses the structure of cyclic semirings with noncommutative addition. In the infinite case, the addition is idempotent and is either left or right. Addition of a finite cyclic semirings can be either idempotent or nonidempotent. In the finite additively idempotent cyclic semiring, addition is reduced to the addition of a cyclic subsemiring with commutative addition and an absorbing element for multiplication and the addition of a cycle that is a finite semifield.

Finite semifields with a large nucleus and higher secant varieties to Segre varieties

Abstract In [Ball, Ebert, Lavrauw, J. Algebra 311: 117–129, 2007] a geometric construction was given of a finite semifield from a certain configuration of two subspaces with respect to a Desarguesian spread in a finite-dimensional vector space over a finite field. Moreover, it was proved that any finite semifield can be obtained in this way. In [Lavrauw, Finite Fields Appl. 14: 897–910, 2008] we proved that the configuration needed for the geometric construction given in [Ball, Ebert, Lavrauw, J. Algebra 311: 117–129, 2007] for finite semifields is equivalent with an (n – 1)-dimensional subspace skew to a determinantal hypersurface in PG(n 2 – 1, q), and provided an answer to the isotopism problem in [Ball, Ebert, Lavrauw, J. Algebra 311: 117–129, 2007]. In this paper we give a generalisation of the BEL-construction using linear sets, and then concentrate on this configuration and the isotopism problem for semifields with nuclei that are larger than its centre.

Fractional dimensions in semifields of odd order

A finite semifield D is considered a fractional dimensional semifield if it contains a subsemifield E such that ? := 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.

Classification of semifields of order 64

A finite semifield D is a finite nonassociative ring with identity such that the set D ∗ = D ∖ { 0 } is closed under the product. In this paper we obtain a computer-assisted description of all semifields of order 64, which completes the classification of finite semifields of order at most 125.

On the isotopism classes of finite semifields

A projective plane is called a translation plane if there exists a line L such that the group of elations with axis L acts transitively on the points not on L. A translation plane whose dual plane is also a translation plane is called a semifield plane. The ternary ring corresponding to a semifield plane can be made into a non-associative algebra called a semifield, and two semifield planes are isomorphic if and only if the corresponding semifields are isotopic. In [S. Ball, G. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (1) (2007) 117–129] it was shown that each finite semifield gives rise to a particular configuration of two subspaces with respect to a Desarguesian spread, called a BEL-configuration, and vice versa that each BEL-configuration gives rise to a semifield. In this manuscript we investigate the question when two BEL-configurations determine isotopic semifields. We show that there is a one-to-one correspondence between the isotopism classes of finite semifields and the orbits of the action a subgroup of index two of the automorphism group of a Segre variety on subspaces of maximum dimension skew to a determinantal hypersurface.

PRIMITIVITY OF FINITE SEMIFIELDS WITH 64 AND 81 ELEMENTS

A finite semifield D is a finite nonassociative ring with identity such that the set D* = D \{0} is a loop under the product. Wene conjectured in [1] that any finite semifield is either right or left primitive, i.e. D* is the set of right (or left) principal powers of an element in D. In this paper we study the primitivity of finite semifields with 64 and 81 elements.

On the Hughes–Kleinfeld and Knuth’s semifields two-dimensional over a weak nucleus

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 ? of a finite field $$\mathbb {K}$$ and elements $$\mu, \eta \in \mathbb{K}$$ with the property that $$x^{\sigma+1}+\mu{x} - \eta$$ has no roots in $$\mathbb{K}$$ . 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 $$(\mathbb{K},\sigma,\mu,\eta)$$ . 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.

A geometric construction of finite semifields

We give a geometric construction of a finite semifield from a certain configuration of two subspaces with respect to a Desarguesian spread in a finite-dimensional vector space over a finite field, and prove that any finite semifield can be obtained in this way. Although no new semifield planes are constructed here, we give explicit subspaces from which some known families of semifields can be constructed. In 1965 Knuth [D.E. Knuth, Finite semifields and projective planes, J. Algebra 2 (1965) 182–217] showed that each finite semifield generates in total six (not necessarily pairwise non-isotopic) semifields. In certain cases, the geometric construction obtained here allows one to construct another six (not necessarily pairwise non-isotopic) semifields, which may or may not be isotopic to any of the six semifields obtained by Knuth's operations. Explicit formulas are calculated for the multiplications of the twelve semifields associated with a semifield that is of rank two over its left nucleus.

Semifield flat flocks

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.

Primitive and Non Primitive Finite Semifields#

A finite semifield (or finite division ring) D is a finite nonassociative ring with identity such that the set is closed under the product and it is a loop. For an arbitrary finite semifield D the structure of its multiplicative loop D* is not known. G.P. Wene introduced the concept of right primitive semifield for those finite semifields D such that D* is equal to the set of principal powers of an element of the semifield, and proved that any semifield of 16, 27, 32, 125 and 343 elements is right primitive. Moreover, he showed that all commutative semifields three-dimensional over a finite field of odd characteristic are right primitive, and conjectured that any finite semifield is right primitive. In this work, we extend his results to any three-dimensional finite semifield over its center, and present a counterexample to the above mentioned conjecture.

The six semifield planes associated with a semifield flock

In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array ( a ijk ) 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+4 xf( 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.

�ber die Projektivit�tengruppen von Semik�rperebenen gerader Ordnung

Let S be a finite proper semifield of even order and II the group of projectivities of the projective plane over S.

Commutative semifields, two dimensional over their middle nuclei

Then we say that S is a (finite) semzj?eld. The subset N, = (n E S: (xn)y = x(ny), Vx, y E S} is known as the middle nucleus of S. N, is a field, and S may be regarded as a (left or right) vector space over N,. A semifield in which multiplication is not associative (i.e., N, # S) is called proper. A finite semifield necessarily has prime power order and proper finite semifields exist for all orders q = p’, p prime, where Y > 3 if p is odd and r>4ifp=2. There is a considerable link between finite semifields and finite projective planes of Lenz-Barlotti class V.1. Indeed, any finite proper semilield gives rise to such a plane, and conversely any plane of the above type yields many isotopic, but not necessarily isomorphic, semifields. The paper by Knuth [4] is an excellent survey of finite semifields and their connections with projective planes. We mention one of the examples given in 141‘