Some varieties of algebraic systems of type ((n),(m))

2019 ◽  
Vol 12 (01) ◽  
pp. 1950005
Author(s):  
D. Phusanga ◽  
J. Koppitz
Keyword(s):  
Algebraic System ◽  
Natural Numbers ◽  
Algebraic Systems ◽  
Term Operation ◽  
Type Formula

In the present paper, we classify varieties of algebraic systems of the type [Formula: see text], for natural numbers [Formula: see text] and [Formula: see text], which are closed under particular derived algebraic systems. If we replace in an algebraic system the [Formula: see text]-ary operation by an [Formula: see text]-ary term operation and the [Formula: see text]-ary relation by the [Formula: see text]-ary relation generated by an [Formula: see text]-ary formula, we obtain a new algebraic system of the same type, which we call derived algebraic system. We shall restrict the replacement to so-called “linear” terms and atomic “linear” formulas, respectively.

scholarly journals Computability and the algebra of fields: Some affine constructions

1980 ◽  
Vol 45 (1) ◽  
pp. 103-120 ◽  
Author(s):  
J. V. Tucker
Keyword(s):  
Algebraic Structure ◽  
Algebraic System ◽  
Finiteness Condition ◽  
Coordinate Systems ◽  
Natural Numbers ◽  
Algebraic Foundation ◽  
Image Position ◽  
Technical Idea ◽  
Natural Way ◽  
General Method

A natural way of studying the computability of an algebraic structure or process is to apply some of the theory of the recursive functions to the algebra under consideration through the manufacture of appropriate coordinate systems from the natural numbers. An algebraic structure A = (A; σ1,…, σk) is computable if it possesses a recursive coordinate system in the following precise sense: associated to A there is a pair (α, Ω) consisting of a recursive set of natural numbers Ω and a surjection α: Ω → A so that (i) the relation defined on Ω by n ≡α m iff α(n) = α(m) in A is recursive, and (ii) each of the operations of A may be effectively followed in Ω, that is, for each (say) r-ary operation σ on A there is an r argument recursive function on Ω which commutes the diagramwherein αr is r-fold α × … × α.This concept of a computable algebraic system is the independent technical idea of M.O.Rabin [18] and A.I.Mal'cev [14]. From these first papers one may learn of the strength and elegance of the general method of coordinatising; note-worthy for us is the fact that computability is a finiteness condition of algebra—an isomorphism invariant possessed of all finite algebraic systems—and that it serves to set upon an algebraic foundation the combinatorial idea that a system can be combinatorially presented and have effectively decidable term or word problem.

n-Cyclic Refined Neutrosophic Algebraic Systems of Sub-Indeterminacies, An Application to Rings and Modules

2020 ◽  
pp. 81-95
Author(s):  
admin admin ◽  
Keyword(s):  
Algebraic System ◽  
Algebraic Systems

Refining the indeterminate I into many levels of indeterminacy is a way to explore many neutrosophic algebraic structures.This paper introduces the concept of n-cyclic refined algebraic system of sub-indeterminacies as a new way to refine a neutrosophic indeterminate I. This idea will be used to introduce the notion of n-cyclic refined neutrosophic ring and to study its AH-substructures. Also, this work presents the concept of n-cyclic refined neutrosophic modules with many related structures.

Diophantine quadruples and near-Diophantine quintuples from P3,K sequences

2017 ◽  
Vol 10 (01) ◽  
pp. 1750010
Author(s):  
A. M. S. Ramasamy
Keyword(s):  
Infinite Number ◽  
Fibonacci Numbers ◽  
Natural Numbers ◽  
Text Type ◽  
Type Formula ◽  
Fourth Term ◽  
Unexplored Area ◽  
Diophantine Quintuples ◽  
Diophantine Quadruples

The question of a non-[Formula: see text]-type [Formula: see text] sequence wherein the fourth term shares the property [Formula: see text] with the first term has not been investigated so far. The present paper seeks to fill up the gap in this unexplored area. Let [Formula: see text] denote the set of all natural numbers and [Formula: see text] the sequence of Fibonacci numbers. Choose two integers [Formula: see text] and [Formula: see text] with [Formula: see text] such that their product increased by [Formula: see text] is a square [Formula: see text]. Certain properties of the sequence [Formula: see text] defined by the relation [Formula: see text] are established in this paper and polynomial expressions for Diophantine quadruples from the [Formula: see text] sequence [Formula: see text] are derived. The concept of a near-Diophantine quintuple is introduced and it is proved that there exist an infinite number of near-Diophantine quintuples.

scholarly journals On Meromorphisms of Algebraic Systems

1966 ◽  
Vol 27 (2) ◽  
pp. 559-569 ◽  
Author(s):  
Junji Hashimoto
Keyword(s):  
Algebraic System ◽  
Algebraic Systems

In the present paper by an algebraic system (algebra) A we shall mean a system with a set F of operations fλ: (x1,…, xn) ∈ A × · · · × A → fλ(x1,…, xn) ∈ A. A polynomial p(x1, …, xr) is a function of variables x1,…, xr which is either one of the xi, or (recursively) a result of some operation fλ(p1,…, pn) performed on other polynomials pi. An algebra A may satisfy a set R of identities p(x1,…, xr) = q(x1,…, xs), and then A shall be called an (F, R)-algebra.

Hypersatisfaction of quantifier free formulas in algebraic systems

2016 ◽  
Vol 09 (03) ◽  
pp. 1650057
Author(s):  
Dara Phusanga ◽  
Jintana Joomwong
Keyword(s):  
Algebraic System ◽  
Solid Model ◽  
Algebraic Systems ◽  
General Algebra

In [Hyperformulas and solid algebraic systems, Studia Logica 90(2) (2008) 263–286], the theory of hyperidentities and solid varieties (see [K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, hyperequational classes, and clone congruences, in Contributions to General Algebra, Vol. 7 (Verlag Hölder-Pichler-Tempsky, Wien, 1991), pp. 97–118]) was extended to algebraic systems and solid model classes of algebraic system. In this paper, we will present a different approach which is based on the concept of the term operations and the realization of quantifier free formulas.

H∞Control of Fractional Nonlinear Differential Systems

2014 ◽  
Vol 945-949 ◽  
pp. 2737-2740
Author(s):  
Li Li
Keyword(s):  
Control Strategy ◽  
Algebraic System ◽  
Structural Characteristics ◽  
Robust Controller ◽  
Differential Systems ◽  
Algebraic Systems ◽  
Differential Algebraic System ◽  
Fractional Differential ◽  
Fractional Control ◽  
Nonlinear Differential Systems

In this paper, the stabilization of nonlinear fractional differential algebraic system is constructed. The fractional nonlinear differential algebraic systems (FNDAS) in presence of disturbance, we construct a fractional control strategy. The proposed stabilization and robust controller effectively take advantage of the structural characteristics of FNDAS and is simple in form.

scholarly journals Numerical semigroups of small and large type

2021 ◽  
pp. 1-20
Author(s):  
Deepesh Singhal
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Natural Numbers ◽  
Type Formula ◽  
Explicit Characterization ◽  
Large Type ◽  
Number Formula ◽  
Genus Formula

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].

scholarly journals ENLARGED MIXED SHIMURA VARIETIES, BI-ALGEBRAIC SYSTEM AND SOME AX TYPE TRANSCENDENTAL RESULTS

2019 ◽  
Vol 7 ◽  
Author(s):  
ZIYANG GAO
Keyword(s):  
Algebraic System ◽  
Shimura Varieties ◽  
Algebraic Systems

We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We study their bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its relation with the logarithmic Ax theorem and the Ax-Lindemann theorem which we shall prove. All these bi-algebraic and transcendental results extend their counterparts for mixed Shimura varieties. In the end we briefly discuss the André–Oort and Zilber–Pink type problems for enlarged mixed Shimura varieties.

scholarly journals On finite approximations of topological algebraic systems

2007 ◽  
Vol 72 (1) ◽  
pp. 1-25 ◽  
Author(s):  
L. Yu. Glebsky ◽  
E. I. Gordon ◽  
C. Ward Henson
Keyword(s):  
Algebraic System ◽  
Nonstandard Analysis ◽  
Space Structures ◽  
Locally Compact ◽  
Algebraic Systems ◽  
Finite Approximations ◽  
Similar Characterization ◽  
Compact Field ◽  
Associative Rings

AbstractWe introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class . If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class of algebraic systems. One characterization of this concept states that A is locally embedded in iff it is a subsystem of an ultraproduct of systems from . In this paper we obtain a similar characterization of approximability of a locally compact system A by systems from using the language of nonstandard analysis.In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula φ holds in A if and only if all precise enough approximations of φ hold in all precise enough approximations of A.We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be non-commutative). Finite approximations of the field ℝ can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings.

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