Soft Union Research Articles

Modified Suture-Button Latarjet Procedure With Coracoacromial Ligament and Pectoralis Minor Preservation Achieves Good Clinical Outcomes at 2-Year Follow-Up: Case Series of Latarjet Technique

To evaluate whether the modified suture-button Latarjet procedure with coracoacromial ligament (CAL) and pectoralis minor (PM) preservation could achieve excellent outcomes at the 2-year follow-up. During January 2019 to January 2021, the data of patients who underwent modified suture-button Latarjet with CAL and PM preservation in our department were collected. The glenoid bone loss of these patients was greater than 20% or greater than 10% with high demands for exercise. Partial coracoid osteotomy was based on the results of a preoperative 3-dimensional computed tomography evaluation of the glenoid defect area and corresponding coracoid process morphology. The preoperative and postoperative clinical results were assessed. The minimal clinically important difference (MCID) was used to compare improvement in clinical outcomes. Graft-glenoid union and remodeling were assessed using postoperative 3-dimensional computed tomography, and magnetic resonance imaging was performed to confirm the integrity of the CAL and PM postoperatively. In total, 35 patients were included in this study; the mean follow-up time was 26.9 ± 1.9 months. No case of recurrent dislocation or sublaxity. Significant improvements were observed in mean visual analog scale (VAS) scores for pain during motion, American Shoulder and Elbow Surgeons (ASES) score, Rowe score, and Walch-Duplay score (P < .001). The percentage of patients achieving at least an MCID improvement in clinical outcomes was VAS 85.71%, ASES 97.14%, Rowe 100%, and Walch-Duplay 97.14%. Thirty-three patients (94.3% of all cases) were able to return to their preoperative sport levels, 34 grafts (97.1%) achieved bone union (1 soft union) in 6.3 ± 2.2 months, and the coracoid grafts restored 97.1 ± 4.0% of the perfect-fitting circle at the last follow-up. Postoperative computed tomography scan showed that 31 grafts (88.6%) were placed ideally in vertical view. In the axial view, 25 grafts (82.9%) were flushed to the glenoid, whereas 1 and 5 grafts were fixed medially and laterally, respectively. The CAL and PM were visualized postoperatively. No arthropathy was observed in any patient at the last follow-up. The modified suture-button Latarjet procedure with CAL and PM preservation obtained good clinical and radiological results without recurrence or complications. A substantial number of patients (>85%) achieved the MCID for the VAS, ASES, Rowe, and Walch-Duplay scores. In addition, the malpositioned graft (17.1%) did not cause arthropathy of the joints at 2-year follow-up. Level IV, retrospective case series.

Finite soft-open sets: characterizations, operators and continuity

&lt;abstract&gt;&lt;p&gt;In this paper, we present a novel family of soft sets named "finite soft-open sets". The purpose of investigating this kind of soft sets is to offer a new tool to structure topological concepts that are stronger than their existing counterparts produced by soft-open sets and their well-known extensions, as well as to provide an environment that preserves some topological characteristics that have been lost in the structures generated by celebrated extensions of soft-open sets, such as the distributive property of a soft union and intersection for soft closure and interior operators, respectively. We delve into a study of the properties of this family and explore its connections with other known generalizations of soft-open sets. We demonstrate that this family strictly lies between the families of soft-clopen and soft-open sets and derive under which conditions they are equivalent. One of the unique features of this family that we introduce is that it constitutes an infra soft topology and fails to be a supra soft topology. Then, we make use of this family to exhibit some operators in soft settings, i.e., soft $ fo $-interior, $ fo $-closure, $ fo $-boundary, and $ fo $-derived. In addition, we formulate three types of soft continuity and look at their main properties and how they behave under decomposition theorems. Transition of these types between realms of soft topologies and classical topologies is examined with the help of counterexamples. On this point, we bring to light the role of extended soft topologies to validate the properties of soft topologies by exploring them for classical topologies and vice-versa.&lt;/p&gt;&lt;/abstract&gt;

Binary Bipolar Soft Points and Topology on Binary Bipolar Soft Sets with Their Symmetric Properties

The aim of this paper is to give an interesting connection between two mathematical approaches to vagueness: binary bipolar soft sets and binary bipolar soft topology. The binary bipolar soft points are defined using binary bipolar soft sets. The binary bipolar soft set will be the binary bipolar soft union of its binary bipolar soft points. Moreover, the notion of binary bipolar soft topological spaces over two universal sets and a parameter set is proposed. Some topological properties of binary bipolar soft sets, such as binary bipolar soft open, binary bipolar soft closed, binary bipolar soft closure, binary bipolar soft interior, and binary bipolar soft boundary, are introduced. Some important properties of these classes of binary bipolar soft sets are investigated. Furthermore, the symmetry relation is compared between binary bipolar soft topology and binary soft topology on a common universe set. Finally, some results and counterexamples are demonstrated to explain this work.

Baire Category Soft Sets and Their Symmetric Local Properties

In this paper, we study soft sets of the first and second Baire categories. The soft sets of the first Baire category are examined to be small soft sets from the point of view of soft topology, while the soft sets of the second Baire category are examined to be large. The family of soft sets of the first Baire category in a soft topological space forms a soft σ-ideal. This contributes to the development of the theory of soft ideal topology. The main properties of these classes of soft sets are discussed. The concepts of soft points where soft sets are of the first or second Baire category are introduced. These types of soft points are subclasses of non-cluster and cluster soft sets. Then, various results on the first and second Baire category soft points are obtained. Among others, the set of all soft points at which a soft set is of the second Baire category is soft regular closed. Moreover, we show that there is symmetry between a soft set that is of the first Baire category and a soft set in which each of its soft points is of the first Baire category. This is equivalent to saying that the union of any collection of soft open sets of the first Baire category is again a soft set of the first Baire category. The last assertion can be regarded as a generalized version of one of the fundamental theorems in topology known as the Banach Category Theorem. Furthermore, it is shown that any soft set can be represented as a disjoint soft union of two soft sets, one of the first Baire category and the other not of the first Baire category at each of its soft points.

Soft Directed Graphs, Some of Their Operations, and Properties

The soft set theory proposed by D. Molodtsov in 1999 is a general mathematical method for dealing with uncertain data. Now many researchers are applying soft set theory in decision making problems. Graph theory is the mathematical study of objects and their pairwise relationships, known as vertices and edges, respectively. The concept of soft graphs is used to provide a parameterized point of view for graphs. Directed graphs can be used to analyze and resolve problems with electrical circuits, project timelines, shortest routes, social links and many other issues. We introduced the notion of the soft directed graph by applying the concepts of soft set in a directed graph. In this paper, we introduce the concept of soft subdigraph and some soft directed graph operations like AND operation, OR operation, soft union, extended union, extended intersection, restricted union and restricted intersection and investigate some of their properties.

Completely weakly, quasi-regular semigroups characterized by soft union Quasi ideals, (generalized) bi-ideals and semiprime ideals

Completely weakly, quasi-regular semigroups characterized by soft union Quasi ideals, (generalized) bi-ideals and semiprime ideals

Two Different Types of Soft Near-Fields

Molodtsov’s soft set is a novel mathematical approach to address models containing uncertainty. The algebraic structures of this set approach are prominent issues. The main objective of this paper is to contribute algebraic structures in the soft set theory. Relatedly, the notions of soft intersection near-field and soft union near-field, and their fundamental properties are introduced. Also, some findings and results related to the emerging soft algebraic structures are included.

On Near-Rings with Soft Union Ideals and Applications

In this paper, we define soft union near-ring on a soft set by using soft sets, inclusion relation and union of sets. This new notion functions as a bridge among soft set theory, set theory and near-ring theory. We then derive its basic properties and investigate the relationship between soft intersection near-ring and soft union near-ring. Furthermore, we obtain some analog of classical near-ring theoretic concepts for soft union near-ring and give the applications of soft union near-ring to near-ring theory.

Soft Semi ω-Open Sets

In this paper, we introduce the class of soft semi ω-open sets of a soft topological space (X,τ,A), using soft ω-open sets. We show that the class of soft semi ω-open sets contains both the soft topology τω and the class of soft semi-open sets. Additionally, we define soft semi ω-closed sets as the class of soft complements of soft semi ω-open sets. We present here a study of the properties of soft semi ω-open sets, especially in (X,τ,A) and (X,τω,A). In particular, we prove that the class of soft semi ω-open sets is closed under arbitrary soft union but not closed under finite soft intersections; we also study the correspondence between the soft topology of soft semi ω-open sets of a soft topological space and their generated topological spaces and vice versa. In addition to these, we introduce the soft semi ω-interior and soft semi ω-closure operators via soft semi ω-open and soft semi ω-closed sets. We prove several equations regarding these two new soft operators. In particular, we prove that these operators can be calculated using other usual soft operators in both of (X,τ,A) and (X,τω,A), and some equations focus on soft anti-locally countable soft topological spaces.

Soft ωp-Open Sets and Soft ωp-Continuity in Soft Topological Spaces

We define soft ωp-openness as a strong form of soft pre-openness. We prove that the class of soft ωp-open sets is closed under soft union and do not form a soft topology, in general. We prove that soft ωp-open sets which are countable are soft open sets, and we prove that soft pre-open sets which are soft ω-open sets are soft ωp-open sets. In addition, we give a decomposition of soft ωp-open sets in terms of soft open sets and soft ω-dense sets. Moreover, we study the correspondence between the soft topology soft ωp-open sets in a soft topological space and its generated topological spaces, and vice versa. In addition to these, we define soft ωp-continuous functions as a new class of soft mappings which lies strictly between the classes of soft continuous functions and soft pre-continuous functions. We introduce several characterizations for soft pre-continuity and soft ωp-continuity. Finally, we study several relationships related to soft ωp-continuity.

T-soft equality relation

The desire of generalizing some set-theoretic properties to the soft set theory motivated many researchers to define various types of soft operators. For example, they redefined the complement of a soft set, and soft union and intersection between two soft sets in a way that satisfies De Morgan's laws. In this paper, we introduce and study the concepts of $T$-soft subset and $T$-soft equality relations. Then, we utilize them to define the concepts of $T$-soft union and $T$-soft intersection for arbitrary family of soft sets. By $T$-soft union, we successfully keep some classical properties via soft set theory. We conclude this work by giving and investigating new types of soft linear equations with respect to some soft equality relations. Illustrative examples are provided to elucidate main obtained results.

Geometrical analysis for assessing torsional alignment of humerus

BackgroundCompared to other types of surgeries, minimally invasive surgeries (MISs) of humeral shaft fractures are associated with less radial nerve injury, less soft tissue injury and higher union rate. However, malrotation often occurs in MISs when closed reduction methods are used.This study aims to define specific palpable landmarks to help surgeons determine the correct torsional angle and reduce the incidence of malrotation.MethodsTwenty-eight normal humeral computed tomography scans were retrieved from our image database. One line was drawn through the vertices of the intertubercular sulcus of the humeral head in the coronal view, and another line was drawn through the longest axis between the medial and lateral condyles in the coronal view. The angle between these two lines was measured at least 3 times for each scan.ResultsThe profile of the intertubercular sulcus tangent line of the humeral head and the axis of the distal humerus was identified as the most accurate method for assessing the precision of torsion during MIS for humeral shaft fractures. The transepicondylar axis line is more internally rotated than the intertubercular sulcus tangent line. The mean angle was measured to be 41.1 degrees.ConclusionsThe axis of the distal humeral condyles is internally rotated by approximately 41.1 degrees compared with the intertubercular sulcus tangent line of the humeral head. Minimally invasive surgeries can be performed by using these palpable landmarks. The torsional deformities can be reduced with the proper angle adjustment without the need for fluoroscopy. It can also be used to treat unstable comminuted humeral fractures.Level of evidenceRetrospective Study, Diagnostic study, Level III.

A NEW APPROACH TO M(G)-GROUP SOFT UNION ACTION AND ITS APPLICATIONS TO M(G)-GROUP THEORY

In this paper, we define a new type of M(G)-group action , called M(G)-group soft union(SU) action and M(G)-ideal soft union(SU) action on a soft set. This new concept illustrates how a soft set effects on an M(G)-group in the mean of union and inclusion of sets and its function as bridge among soft set theory, set theory and M(G)-group theory. We also obtain some analog of classical M(G)- group theoretic concepts for M(G)-group SU-action. Finally, we give the application of SU-actions on M(G)-group to M(G)-group theory.

Decomposition of supra soft locally closed sets and supra SLC-continuity

In this paper, we introduce two different notions of generalized supra soft sets namely supra A--soft sets and supra soft locally closed sets in supra soft topological spaces, which are weak forms of supra open soft sets and discuss their relationships with each other and other supra open soft sets [{it International Journal of Mathematical Trends and Technology} (IJMTT), (2014) Vol. 9 (1):37--56] like supra semi open soft sets, supra pre open soft sets, supra $alpha$--open sets and supra $beta$--open sets. Furthermore, the soft union and intersection of two supra soft locally closed sets have been obtained. We also introduce two different notions of generalized supra soft continuity namely supra soft A--continuous functions and supra SLC--continuous functions. Finally, we obtain decompositions of supra soft continuity: $f_{pu}$ is a supra soft A--continuous if it is both supra soft semi-continuous and supra SLC--continuous, and also $f_{pu}$ is a supra soft continuous if and only if it is both supra soft pre--continuous and supra SLC--continuous. Several examples are provided to illustrate the behavior of these new classes of supra soft sets and supra soft functions.

Soft Uni-Abel-Grassmann's Groups

In this paper, we define soft union AG-group (abbreviated as soft uni-AG-group).We also define e-set and a-inclusion of soft uni-AG-groups, normal soft uni-AG-subgroups, conjugate of soft uni-AG-groups and commutators of AG-groups. We investigate various properties of these notions and provide a variety of relevant examples that are produced by a computer package GAP to illustrate these notations completely.

A new view on AG-groupoid theory via soft sets for uncertainty modeling

In this paper, the concepts soft union AG-groupoids, soft union left (right, two-sided) ideals, (generalized) bi-ideals, interior and quasi-ideals in AG-groupoids are introduced with many illustrating examples and their properties and interrelations are given. Moreover, regular, intra-regular, completely regular, weakly regular and quasi-regular AG-groupoids are characterized by the properties of these soft union ideals.

Soft union interior ideals, quasi-ideals and generalized bi-ideals of rings

In this paper, soft union interior ideals, quasi-ideals and generalized bi-ideals of rings are defined and their properties are obtained and the interrelations of them are given. Moreover regular, regular duo, intra-regular and strongly regular rings are characterized in terms of these soft union ideals. This paper is a following study of [19].

Regular and intra-regular semihypergroups in terms of soft union hyperideals

Regular and intra-regular semihypergroups in terms of soft union hyperideals

A New Approach to M(G)-Group Soft Union Action and its Applications to M(G)-Group Theory

In this paper, we define a new type of M(G)-group action , called M(G)-group soft union(SU) action and M(G)-ideal soft union(SU) action on a soft set. This new concept illustrates how a soft set effects on an M(G)-group in the mean of union and inclusion of sets and its function as bridge among soft set theory, set theory and M(G)-group theory. We also obtain some analog of classical M(G)- group theoretic concepts for M(G)-group SU-action. Finally, we give the application of SU-actions on M(G)-group to M(G)-group theory.

Generalized operations in soft set theory via relaxed conditions on parameters

Soft set theory has been evolved as a very useful mathematical tool to handle uncertainty and ambiguity associated with the real world data based structures. Parameters with certain conditions have been used to classify the data with the help of suitable functions. The aim of this paper is to relax conditions on parameters which lead us to propose some new concepts that consequently generalize existing comparable notions. We introduce the concepts of generalized finite soft equality (gf-soft equality), generalized finite soft union (gf-soft union) and generalized finite soft intersection (gf-soft intersection) of two soft sets. We prove results involving operations introduced herein. Moreover, with the help of examples, it is shown that these operations are proper generalizations of existing comparable operations.