scholarly journals Some relations and subsets generated by principal consistent subset of semigroup with apartness

2002 ◽  
pp. 7-25 ◽  
Author(s):  
Abraham Romano
Keyword(s):  
Set Theory ◽  
Semigroup Theory ◽  
Left Zero ◽  
Constructive Mathematics ◽  
Algebraic Structures ◽  
Main Subject ◽  
Special Cases ◽  
Constructive Algebra ◽  
Consistent Subset ◽  
Semigroup With Apartness

The investigation is in the Constructive algebra in the sense of E. Bishop, F. Richman, W. Ruitenburg, D. van Dalen and A. S. Troelstra. Algebraic structures with apartness the first were defined and studied by A. Heyting. After that, some authors studied algebraic structures in constructive mathematics as for example: D. van Dalen, E. Bishop, P. T. Johnstone, A. Heyting, R. Mines, J. C. Mulvey, F. Richman, D. A. Romano, W. Ruitenburg and A. Troelstra. This paper is one of articles in their the author tries to investigate semugroups with apartnesses. Relation q on S is a coequality relation on S if it is consistent, symmetric and cotran-sitive; coequality relation is generalization of apatness. The main subject of this consideration are characterizations of some coequality relations on semigroup S with apartness by means od special ideals J(a) = {x E S : a# SxS}, principal consistent subsets C(a) = {x E S : x# SaS} (a E S) of S and by filled product of relations on S. Let S = (S, =, 1) be a semigroup with apartness. As preliminaries we will introduce some special notions, notations and results in set theory, commutative ring theory and semigroup theory in constructive mathematics and we will give proofs of several general theorems in semigroup theory. In the next section we will introduce relation s on S by (x, y) E s iff y E C(x) and we will describe internal filfulments c(s U s?1) and c(s ? s?1) and their classes A(a) = ?An(a) and K(a) = ?Kn(a) respectively. We will give the proof that the set K(a) is maximal strongly extensional consistent ideal of S for every a in S. Before that, we will analyze semigroup S with relation q = c(s U s?1 ) in two special cases: (i) the relation q is a band coequality relation on S : (ii) q is left zero band coequality relation on S. Beside that, we will introduce several compatible equality and coequality relations on S by sets A(a), An(a), K(a) and Kn(a).

On soft set-valued maps and integral inclusions

2021 ◽  
pp. 1-15
Author(s):  
Monairah Alansari ◽  
Shehu Shagari Mohammed ◽  
Akbar Azam
Keyword(s):  
Fixed Point ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Fixed Point Theorems ◽  
Soft Set ◽  
Mathematical Tool ◽  
Multivalued Maps ◽  
Soft Sets ◽  
Special Cases ◽  
New Development

As an improvement of fuzzy set theory, the notion of soft set was initiated as a general mathematical tool for handling phenomena with nonstatistical uncertainties. Recently, a novel idea of set-valued maps whose range set lies in a family of soft sets was inaugurated as a significant refinement of fuzzy mappings and classical multifunctions as well as their corresponding fixed point theorems. Following this new development, in this paper, the concepts of e-continuity and E-continuity of soft set-valued maps and αe-admissibility for a pair of such maps are introduced. Thereafter, we present some generalized quasi-contractions and prove the existence of e-soft fixed points of a pair of the newly defined non-crisp multivalued maps. The hypotheses and usability of these results are supported by nontrivial examples and applications to a system of integral inclusions. The established concepts herein complement several fixed point theorems in the framework of point-to-set-valued maps in the comparable literature. A few of these special cases of our results are highlighted and discussed.

scholarly journals Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 321 ◽  
Author(s):  
Mehmet Çelik ◽  
Moges Shalla ◽  
Necati Olgun
Keyword(s):  
Group Theory ◽  
Significant Role ◽  
Classical Group ◽  
Algebraic Structures ◽  
Algebraic Systems ◽  
Homomorphism Theorem ◽  
Special Cases ◽  
Isomorphism Theorems

In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related.

scholarly journals Final coalgebras as greatest fixed points in ZF set theory

1999 ◽  
Vol 9 (5) ◽  
pp. 545-567 ◽  
Author(s):  
LAWRENCE C. PAULSON
Keyword(s):  
Fixed Points ◽  
Set Theory ◽  
Theorem Prover ◽  
Infinite Length ◽  
Special Cases ◽  
Definition Of ◽  
Recursive Definitions ◽  
Ordered Pairs

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo–Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's solution and substitution lemmas are proved in the style of Rutten and Turi (1993). The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint.Compared with previous work (Paulson, 1995a), iterated substitutions and solutions are considered, as well as final coalgebras defined with respect to parameters. The disjoint sum construction is replaced by a smoother treatment of urelements that simplifies many of the derivations.The theory facilitates machine implementation of recursive definitions by letting both inductive and coinductive definitions be represented as fixed points. It has already been applied to the theorem prover Isabelle (Paulson, 1994).

Stewart Shapiro. Introduction—intensional mathematics and constructive mathematics. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, vol. 113, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 1–10. - Stewart Shapiro. Epistemic and intuitionistic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 11–46. - John Myhill. Intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 47–61. - Nicolas D. Goodman. A genuinely intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 63–79. - Andrej Ščedrov. Extending Godel's modal interpretation to type theory and set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 81–119. - Robert C. Flagg. Church's thesis is consistent with epistemic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 121–172.

1991 ◽  
Vol 56 (4) ◽  
pp. 1496-1499 ◽  
Author(s):  
Craig A. Smoryński
Keyword(s):  
New York ◽  
Set Theory ◽  
Type Theory ◽  
Foundations Of Mathematics ◽  
Constructive Mathematics ◽  
Modal Interpretation ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Epistemic Arithmetic ◽  
Intuitionistic Arithmetic

scholarly journals Uniformly supported sets and fixed points properties

2020 ◽  
Vol 36 (3) ◽  
pp. 351-364
Author(s):  
ANDREI ALEXANDRU ◽  
GABRIEL CIOBANU
Keyword(s):  
Fixed Points ◽  
Set Theory ◽  
Algebraic Structures

The theory of finitely supported algebraic structures is a reformulation of Zermelo-Fraenkel set theory in which every set-based construction is finitely supported according to a canonical action of a group of permutations of some basic elements named atoms. In this paper we study the properties of finitely supported sets that contain infinite uniformly supported subsets, as well as the properties of finitely supported sets that do not contain infinite uniformly supported subsets. Particularly, we focus on fixed points properties.

scholarly journals A Soft Interval Based Decision Making Method and Its Computer Application

2021 ◽  
Vol 46 (3) ◽  
pp. 273-296
Author(s):  
Gözde Yaylalı ◽  
Nazan Çakmak Polat ◽  
Bekir Tanay
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Soft Set ◽  
Computer Application ◽  
Complex Data ◽  
Soft Sets ◽  
Mathematical Algorithm ◽  
Huge Data ◽  
Special Cases ◽  
General Method

Abstract In today’s society, decision making is becoming more important and complicated with increasing and complex data. Decision making by using soft set theory, herein, we firstly report the comparison of soft intervals (SI) as the generalization of interval soft sets (ISS). The results showed that SIs are more effective and more general than the ISSs, for solving decision making problems due to allowing the ranking of parameters. Tabular form of SIs were used to construct a mathematical algorithm to make a decision for problems that involves uncertainties. Since these kinds of problems have huge data, constructing new and effective methods solving these problems and transforming them into the machine learning methods is very important. An important advance of our presented method is being a more general method than the Decision-Making methods based on special situations of soft set theory. The presented method in this study can be used for all of them, while the others can only work in special cases. The structures obtained from the results of soft intervals were subjected to test with examples. The designed algorithm was written in recently used functional programing language C# and applied to the problems that have been published in earlier studies. This is a pioneering study, where this type of mathematical algorithm was converted into a code and applied successfully.

scholarly journals On co-Filters in Semigroups with Apartness

2021 ◽  
Vol 45 (4) ◽  
pp. 607-613
Author(s):  
DANIEL A. ROMANO ◽  
Keyword(s):  
Intuitionistic Logic ◽  
Constructive Mathematics ◽  
Ordered Semigroup ◽  
Philosophical Orientation ◽  
Semigroup With Apartness ◽  
Bishop’S Constructive Mathematics

The logical environment of this research is the Intuitionistic Logic and principled-philosophical orientation of the Bishop’s Constructive Mathematics. In this paper, basing our consideration on the sets with the apartness relation, we analyze the lattices of all co-filters of an ordered semigroup under a co-quasiorder as a continuation of our article [?]. We prove a number of results related to co-filters in a semigroup with apartness and the lattice of all co-filters of such semigroups.

scholarly journals Asteroidal Motion at Commensurabilities Treated in Three Dimensions

1979 ◽  
Vol 81 ◽  
pp. 207-215 ◽  
Author(s):  
Joachim Schubart
Keyword(s):  
Three Dimensional ◽  
Three Dimensions ◽  
Main Subject ◽  
Long Period ◽  
Period Effects ◽  
Special Cases ◽  
The Mean ◽  
Work Done ◽  
Asteroidal Motion

This paper consists of a review about work done on three-dimensional motion at commensurabilities of either the mean motions, or of secular periods, and of a report on the author's recent results on some special cases. Real and fictitious asteroidal orbits and the corresponding long-period effects are the main subject of interest. At first, methods are listed.

Algebraic Structure of Step Traces and Interval Traces

2020 ◽  
Vol 175 (1-4) ◽  
pp. 253-280
Author(s):  
Ryszard Janicki ◽  
Łukasz Mikulski
Keyword(s):  
Algebraic Structure ◽  
Formal Analysis ◽  
Concurrent Systems ◽  
Interval Order ◽  
Canonical Forms ◽  
Algebraic Structures ◽  
Main Subject

Traces and their extensions as comtraces, step traces and interval traces are quotient monoids over sequences or step sequences that play an important role in the formal analysis and verification of concurrent systems. Step traces are generalizations of comtraces and classical traces while interval traces are specialized traces that can deal with interval order semantics. The algebraic structures and their properties as projections, hidings, canonical forms and other invariants are very well established for traces and fairly well established for comtraces. For step traces and interval traces they are the main subject of this paper.

scholarly journals Diabetes Disease Analysis Using Rough Soft Set

2018 ◽  
Vol 7 (4.10) ◽  
pp. 316
Author(s):  
M. Manimaran ◽  
B. Praba ◽  
G. Deepa ◽  
V. M. Chandrasekaran ◽  
Krishnamoorthy Venkatesan
Keyword(s):  
Diabetes Mellitus ◽  
Set Theory ◽  
Rough Set ◽  
Soft Set ◽  
Age Group ◽  
Health Issues ◽  
Soft Decision ◽  
High Blood Glucose ◽  
Special Cases ◽  
Disease Analysis

Diabetes is a noteworthy medical issue in both modern and creating nations, and its frequency is rising apparently. It is a metabolic disease in which the person who has been affected will have high blood glucose or high blood sugar. It is mainly because of inadequate production of insulin or the body’s cells do not respond to insulin. In some special cases it may be due to both the reasons. This disease causes a lot of health issues in humans’ life. Rough set and soft set theory plays a major role for dealing with uncertainty and it has been applied in many fields. In this paper we aim at finding the age group of people in which maximum diabetes mellitus occurs using the concept of rough soft set and rough soft decision set.  

Sign in / Sign up

Export Citation Format

Share Document