scholarly journals The Fundamental Theorems of Hyper-Operations

2019 ◽  
Author(s):  
Renny Barrett
Keyword(s):  
Fundamental Theorem ◽  
Natural Numbers ◽  
Higher Rank ◽  
Fundamental Theorem Of Arithmetic ◽  
Fundamental Theorems

We examine the extensions of the basic arithmetical operations of addition and multiplication on the natural numbers into higher-rank hyper-operations also on the natural numbers. We go on to define the concepts of prime and composite numbers under these hyper-operations and derive some results about factorisation, resulting in fundamental theorems analogous to the Fundamental Theorem of Arithmetic.

THE GREATEST COMMON DIVISOR AND OTHER TRIANGULAR NORMS ON THE EXTENDED SET OF NATURAL NUMBERS

2009 ◽  
Vol 17 (01) ◽  
pp. 35-45
Author(s):  
GASPAR MAYOR ◽  
JAUME MONREAL
Keyword(s):  
Direct Product ◽  
Fundamental Theorem ◽  
Greatest Common Divisor ◽  
Natural Numbers ◽  
Prime Decomposition ◽  
Triangular Norms ◽  
Complete Chain ◽  
Fundamental Theorem Of Arithmetic

This paper deals with triangular norms and conorms defined on the extended set N of natural numbers ordered by divisibility. From the fundamental theorem of arithmetic, N can be identified with a lattice of functions from the set of primes to the complete chain {0, 1, 2, …, +∞}, thus our knowledge about (divisible) t-norms on this chain can be applied to the study of t-norms on N. A characterization of those t-norms on N which are a direct product of t-norms on {0, 1, 2, …, +∞} is given and, after introducing the concept of T-prime (prime with respect to a t-norm T), a theorem about the existence of a T-prime decomposition is obtained. This result generalizes the fundamental theorem of arithmetic.

1. Numbers and algebra

2015 ◽  
pp. 1-10
Author(s):  
Peter M. Higgins
Keyword(s):  
Fundamental Theorem ◽  
Common Factor ◽  
Number System ◽  
Rational Numbers ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Natural Numbers ◽  
Division Algorithm ◽  
Fundamental Theorem Of Arithmetic ◽  
And Algebra

‘Numbers and algebra’ introduces the number system and explains several terms used in algebra, including natural numbers, positive and negative integers, rational numbers, number factorization, the Fundamental Theorem of Arithmetic, Euclid’s Lemma, the Division Algorithm, and the Euclidean Algorithm. It proves that any common factor c of a and b is also a factor of any number of the form ax + by, and since the greatest common divisor (gcd) of a and b has this form, which may be found by reversing the steps of the Euclidean Algorithm, it follows that any common factor c of a and b divides their gcd d.

scholarly journals THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

2019 ◽  
pp. 1-25 ◽  
Author(s):  
G. I. LEHRER ◽  
R. B. ZHANG
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorem ◽  
Brauer Algebra ◽  
Tensor Space ◽  
General Linear Supergroup ◽  
Special Cases ◽  
Linear Description ◽  
Tensor Representations ◽  
Capelli Identities ◽  
Fundamental Theorems

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$ . This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$ .

The Fundamental Theorem of Arithmetic Dissected

1997 ◽  
Vol 81 (490) ◽  
pp. 53 ◽  
Author(s):  
Ahmet G. Agargun ◽  
Colin R. Fletcher
Keyword(s):  
Fundamental Theorem ◽  
Fundamental Theorem Of Arithmetic

Number theory and other reminiscences of Viscount Cherwell

1988 ◽  
Vol 42 (2) ◽  
pp. 197-204 ◽  
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Fundamental Theorem ◽  
Prime Numbers ◽  
Christ Church ◽  
Active Interest ◽  
Fundamental Theorem Of Arithmetic ◽  
Elegant Proof ◽  
Theory Of Numbers

Lord Cherwell (i) was, of course, a very distinguished ex-perimental physicist but he had (like many others) a considerable active interest in the theory of numbers. I met him in 1930 when Christ Church, Oxford, elected me to a Senior (postgraduate) Scholarship and I migrated there from my original college. Cherwell’s first published work (2) in the theory of numbers was a very simple and elegant proof of the fundamental theorem of arithmetic, that any positive integer can be expressed as a product of prime numbers in just one way (apart from a possible rearrangement of the order of the factors). (A prime is a positive integer greater than 1 whose only factors are 1 and itself.) His proof is by the method of descent (used by Fermat, but not for this problem). Assume the fundamental theorem false and call any number that can be expressed as a product of primes in two or more ways abnormal.

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