Related Experiment Video
Updated: Apr 20, 2026
11:49
Synthesis of Soft Polysiloxane-urea Elastomers for Intraocular Lens Application
Published on: March 8, 2019
13.4K
Concave soft sets, critical soft points, and union-soft ideals of ordered semigroups
Young Bae Jun1, Seok Zun Song2, G Muhiuddin3
1Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Republic of Korea.
Thescientificworldjournal
|November 19, 2014
Summary
This study introduces union-soft semigroups and their ideals, exploring their properties and characterizations. It also examines union-soft products and semiprime soft sets, providing conditions for regular ordered semigroups.
Area of Science:
- Abstract algebra
- Fuzzy set theory
- Soft set theory
Background:
- Soft set theory offers a generalized framework for dealing with uncertainty.
- Union-soft sets extend these capabilities, enabling more nuanced analysis.
- Ideals are fundamental substructures in algebraic systems.
Purpose of the Study:
- Introduce and investigate union-soft semigroups, union-soft l-ideals, and union-soft r-ideals.
- Characterize these union-soft structures.
- Explore union-soft products, semiprime soft sets, and their relation to ideals.
- Establish conditions for regularity in ordered semigroups using union-soft ideals.
Main Methods:
- Definition and exploration of union-soft algebraic structures.
- Development of characterization theorems for union-soft semigroups and ideals.
- Investigation of properties of union-soft products and semiprime soft sets.
- Application of union-soft ideals to determine regularity conditions in ordered semigroups.
Main Results:
- The paper introduces novel concepts: union-soft semigroups, union-soft l-ideals, and union-soft r-ideals.
- Characterizations for these union-soft structures are established.
- Properties of union-soft products and semiprime soft sets are analyzed in relation to union-soft ideals.
- Conditions for regular ordered semigroups are derived using the developed union-soft ideal theory.
- New concepts of concave soft sets and critical soft points are introduced and discussed.
Conclusions:
- The study successfully extends algebraic concepts using union-soft set theory.
- The introduced union-soft ideals provide a powerful tool for analyzing algebraic structures.
- The findings contribute to the understanding of soft algebraic systems and their applications.
Related Concept Videos
01:27Slant Asymptotes
239
A function's behavior is often guided by asymptotic constraints, where one term dominates another, defining a limiting trend. In the given scenario, the mathematical pattern follows a rational function: a cubic term in the numerator is divided by a squared term in the denominator. This results in a function with distinct characteristics, including an oblique asymptote, critical points, and undefined regions.The function's validity is determined by the denominator, which must be nonzero. This...
239
01:30Fundamental Theorem of Algebra
468
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as: with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
468
01:15Routh-Hurwitz Criterion I
713
Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.
To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the...
To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the...
713
01:19Routh-Hurwitz Criterion II
1.3K
In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
1.3K
01:18Singularity Functions for Bending Moment
672
Singularity functions simplify the representation of bending moments in beams subjected to discontinuous loading, allowing the use of a single mathematical expression. For a supported beam AB, with uniform loading from its midpoint M to the right side end B, the approach involves conceptual 'cuts' at specific points to determine the bending moment in each segment. By cutting the beam at a point between A and M, the bending moment for the segment before reaching midpoint M is represented using a...
672
01:29Fundamental Theorem of Calculus II
353
In calculus, the computation of the area under a continuous curve has been fundamentally simplified by applying the Fundamental Theorem of Calculus, Part 2. Rather than relying on the limiting process of summing infinitely many infinitesimal rectangles, this theorem permits direct evaluation using antiderivatives, thereby streamlining the process of definite integration.The Fundamental Theorem of Calculus, Part 2, states that if a function f(x) is continuous on a closed interval [a, b], then...
353