In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity and every non-zero element has a multiplicative inverse.
A near-field is a set together with two binary operations, (addition) and (multiplication), satisfying the following axioms:
A1: is an abelian group.
A2: = for all elements , , of (The associative law for multiplication).
A3: for all elements , , of (The right distributive law).
A4: contains an element 1 such that for every element of (Multiplicative identity).
A5: For every non-zero element of there exists an element such that (Multiplicative inverse).
The above is, strictly speaking, a definition of a right near-field. By replacing A3 by the left distributive law we get a left near-field instead. Most commonly, "near-field" is taken as meaning "right near-field", but this is not a universal convention.
A (right) near-field is called "planar" if it is also a right quasifield. Every finite near-field is planar, but infinite near-fields need not be.
It is not necessary to specify that the additive group is abelian, as this follows from the other axioms, as proved by B.H. Neumann and J.L. Zemmer. However, the proof is quite difficult, and it is more convenient to include this in the axioms so that progress with establishing the properties of near-fields can start more rapidly.
Sometimes a list of axioms is given in which A4 and A5 are replaced by the following single statement:
A4*: The non-zero elements form a group under multiplication.
However, this alternative definition includes one exceptional structure of order 2 which fails to satisfy various basic theorems (such as for all ). Thus it is much more convenient, and more usual, to use the axioms in the form given above. The difference is that A4 requires 1 to be an identity for all elements, A4* only for non-zero elements.
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In mathematics, a quasifield is an algebraic structure where and are binary operations on , much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields. A quasifield is a structure, where and are binary operations on , satisfying these axioms: is a group is a loop, where (left distributivity) has exactly one solution for , Strictly speaking, this is the definition of a left quasifield. A right quasifield is similarly defined, but satisfies right distributivity instead.
En mathématiques, plus précisément en arithmétique et en algèbre générale, la distributivité d'une opération par rapport à une autre est une généralisation de la propriété élémentaire : « le produit d'une somme est égal à la somme des produits ». Par exemple, dans l'expression 2 × (5 + 3) = (2×5) + (2×3), le facteur 2 est distribué à chacun des deux termes de la somme 5 + 3. L'égalité est alors bien vérifiée : à gauche 2 × 8 = 16, à droite 10 + 6 = 16.
We prove a Szemeredi-Trotter type theorem and a sum product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and Sarkozy on the solvability of th ...
Academic Press Inc Elsevier Science2017
In this thesis we study a number of problems in Discrete Combinatorial Geometry in finite spaces. The contents in this thesis are structured as follows: In Chapter 1 we will state the main results and the notations which will be used throughout the thesis. ...