Mathematical Logic Quarterly最新文献

{"title":"On The (k,s)-Hilfer Fractional Derivatives via Wirtinger-Type Inequalities and Their Analytical Applications","authors":"Muhammad Samraiz,&nbsp;Areej Yussouf,&nbsp;Saima Naheed","doi":"10.1002/malq.70020","DOIUrl":"https://doi.org/10.1002/malq.70020","url":null,"abstract":"<div>\u0000 \u0000 <p>This article introduces a novel definition of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>,</mo>\u0000 <mi>s</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(k,s)$</annotation>\u0000 </semantics></math>-Hilfer fractional derivative. Based on this derivative, some fractional Wirtinger type inequalities are established for the <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$L_{p}$</annotation>\u0000 </semantics></math> spaces, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$p &gt; 1$</annotation>\u0000 </semantics></math> by using Hölder's inequality. Various related special cases are also presented. To validate our main results, examples with graphical representations are provided. Applications of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>,</mo>\u0000 <mi>s</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(k,s)$</annotation>\u0000 </semantics></math>-Hilfer fractional Wirtinger-type inequalities are demonstrated in terms of arithmetic mean and geometric mean-type inequality.</p></div>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 2","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147687008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Monotone Infinitary Operation on Ordinals","authors":"Paolo Lipparini","doi":"10.1002/malq.70019","DOIUrl":"https://doi.org/10.1002/malq.70019","url":null,"abstract":"<div>\u0000 \u0000 <p>We introduce an <span></span><math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$ omega$</annotation>\u0000 </semantics></math>-ary operation on the class of the ordinals, which is strictly monotone in all significant cases. We provide order-theoretical characterizations as the rank of a sequence in a well-founded order and as a mixed sum of the ordinals in the sequence.</p></div>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 2","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147614996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Right-Continuous Mappings and Fuzzy Ideals","authors":"Takashi Kuraoka","doi":"10.1002/malq.70018","DOIUrl":"https://doi.org/10.1002/malq.70018","url":null,"abstract":"<div>\u0000 \u0000 <p>We introduce the concept of right-continuous mappings from [0,1) to the power set of finite elements in a complete semilattice and define a closure operator on the set <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>P</mi>\u0000 <mi>r</mi>\u0000 </msub>\u0000 <annotation>$P_r$</annotation>\u0000 </semantics></math> of right-continuous mappings. We show that the lattice of fuzzy ideals on the semilattice of finite elements is isomorphic to the lattice of closed elements concerning the closure operator. Furthermore, we give equivalent conditions for regular closure operators on the set of right-continuous mappings to be quasi-algebraic. Finally, we prove the lattice of fuzzy ideals on the set of finite elements in an algebraic semilattice is subdirectly embedded into the product of copies of the semilattice, and show the subdirect embedding has a universal mapping property.</p></div>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 2","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147567827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Graphical Insights and Applications of Fractional Minkowski and Fejér-Hermite-Hadamard Type Inequalities","authors":"Zeeshan Anwar,&nbsp;Saima Naheed,&nbsp;Ahsan Mehmood,&nbsp;Muhammad Samraiz","doi":"10.1002/malq.70015","DOIUrl":"https://doi.org/10.1002/malq.70015","url":null,"abstract":"<div>\u0000 \u0000 <p>In this study, the Minkowski and Fejér-Hermite-Hadamard (H-H) type inequalities are generalized by utilizing the modified Atangana-Baleanu (A-B) fractional operators. These fractional operators, defined by their nonlocal and nonsingular kernels provide a new way to generalize these classical inequalities. The inequalities are verified through several illustrative examples and corresponding graphs. A new application involving the digamma function is presented to demonstrate the significance of the results. This research opens new avenues for establishing further inequalities via fractional operators.</p></div>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 2","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147567233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Structures of Sign-Boundary and Diagonal Vacancy-Type Standard Contradictions","authors":"Xingxing He,&nbsp;Lan Pan,&nbsp;Yingfang Li,&nbsp;Jun Liu,&nbsp;Luis Martínez","doi":"10.1002/malq.70010","DOIUrl":"https://doi.org/10.1002/malq.70010","url":null,"abstract":"<div>\u0000 \u0000 <p>Automated deduction based on contradiction separation extends the binary resolution principle, offering a novel approach to deductive inference rules. Constructing standard contradictions is essential for its efficiency. This paper systematically investigates two new types of standard contradictions in propositional and first-order logic, enriching the library of standard contradictions and enhancing its effectiveness. First, we define two types of standard contradictions: sign-boundary contradictions and diagonal vacancy-type contradictions. Next, we propose the corresponding construction methods and present their properties related to contradiction composition and literal addition. Furthermore, we explore the transformations between these two types of contradictions and analyze the conditions necessary to construct standard contradictions. Finally, we extend these findings to first-order logic, demonstrating their applicability in more complex logical systems.</p></div>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 2","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147565794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Triangular Norms on Partially Ordered Sets","authors":"Yong Su,&nbsp;Feng Tian,&nbsp;Wenwen Zong,&nbsp;Radko Mesiar","doi":"10.1002/malq.70014","DOIUrl":"https://doi.org/10.1002/malq.70014","url":null,"abstract":"<div>\u0000 \u0000 <p>In this study, we extend the additively generated triangular norms from the framework of the unit interval to that of partially ordered sets. We present several conditions under which this formula <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 <mi>T</mi>\u0000 <mo>(</mo>\u0000 <mi>φ</mi>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>φ</mi>\u0000 <mi>y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$psi T(varphi x, varphi y)$</annotation>\u0000 </semantics></math> yields a t-norm, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mo>,</mo>\u0000 <mi>Q</mi>\u0000 </mrow>\u0000 <annotation>$P, Q$</annotation>\u0000 </semantics></math> are partially ordered sets, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>φ</mi>\u0000 <mo>:</mo>\u0000 <mi>P</mi>\u0000 <mo>→</mo>\u0000 <mi>Q</mi>\u0000 </mrow>\u0000 <annotation>$varphi: P rightarrow Q$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 <mo>:</mo>\u0000 <mi>Q</mi>\u0000 <mo>→</mo>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation>$psi: Q rightarrow P$</annotation>\u0000 </semantics></math> are monotone functions and <span></span><math>\u0000 <semantics>\u0000 <mi>T</mi>\u0000 <annotation>$T$</annotation>\u0000 </semantics></math> is a t-norm/t-conorm on <span></span><math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>$Q$</annotation>\u0000 </semantics></math>. The partially ordered semigroups induced by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 <mi>T</mi>\u0000 <mo>(</mo>\u0000 <mi>φ</mi>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>φ</mi>\u0000 <mi>y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$psi T(varphi x, varphi y)$</annotation>\u0000 </semantics></math> are order-preserving/order-reversing homomorphic to semigroup deformations of the semigroup induced by <span></span><math>\u0000 <semantics>\u0000 <mi>T</mi>\u0000 <annotation>$T$</annotation>\u0000 </semantics></math>.</p></div>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 2","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147564600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ideal Analytic Sets","authors":"Łukasz Mazurkiewicz,&nbsp;Szymon Żeberski","doi":"10.1002/malq.70012","DOIUrl":"10.1002/malq.70012","url":null,"abstract":"<div>\u0000 \u0000 <p>The aim of this study is to give natural examples of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$operatorname{mathbf {Sigma }_1^1}$</annotation>\u0000 </semantics></math>-complete and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Π</mi>\u0000 <mn>1</mn>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$operatorname{mathbf {Pi }_1^1}$</annotation>\u0000 </semantics></math>-complete sets.</p>\u0000 <p>In the first part, we consider ideals on <span></span><math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$omega$</annotation>\u0000 </semantics></math>. We use a unified approach introduced in [4] to create reductions of the collection of ill-founded trees to the ideals, proving <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$operatorname{mathbf {Sigma }_1^1}$</annotation>\u0000 </semantics></math>-completeness of the ideals.</p>\u0000 <p>In the second part, we show the connection between this topic, families of trees and coding of <span></span><math>\u0000 <semantics>\u0000 <mi>σ</mi>\u0000 <annotation>$sigma$</annotation>\u0000 </semantics></math>-ideals of Polish spaces. In particular, we use the unified approach to prove that sets of codes for closed Ramsey-null sets, for closed <span></span><math>\u0000 <semantics>\u0000 <mi>σ</mi>\u0000 <annotation>$sigma$</annotation>\u0000 </semantics></math>-compact sets and for closed not strongly dominating sets are <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Π</mi>\u0000 <mn>1</mn>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$operatorname{mathbf {Pi }_1^1}$</annotation>\u0000 </semantics></math>-complete.</p></div>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147569942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ideal Analytic Sets","authors":"Łukasz Mazurkiewicz,&nbsp;Szymon Żeberski","doi":"10.1002/malq.70012","DOIUrl":"https://doi.org/10.1002/malq.70012","url":null,"abstract":"<div>\u0000 \u0000 <p>The aim of this study is to give natural examples of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$operatorname{mathbf {Sigma }_1^1}$</annotation>\u0000 </semantics></math>-complete and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Π</mi>\u0000 <mn>1</mn>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$operatorname{mathbf {Pi }_1^1}$</annotation>\u0000 </semantics></math>-complete sets.</p>\u0000 <p>In the first part, we consider ideals on <span></span><math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$omega$</annotation>\u0000 </semantics></math>. We use a unified approach introduced in [4] to create reductions of the collection of ill-founded trees to the ideals, proving <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$operatorname{mathbf {Sigma }_1^1}$</annotation>\u0000 </semantics></math>-completeness of the ideals.</p>\u0000 <p>In the second part, we show the connection between this topic, families of trees and coding of <span></span><math>\u0000 <semantics>\u0000 <mi>σ</mi>\u0000 <annotation>$sigma$</annotation>\u0000 </semantics></math>-ideals of Polish spaces. In particular, we use the unified approach to prove that sets of codes for closed Ramsey-null sets, for closed <span></span><math>\u0000 <semantics>\u0000 <mi>σ</mi>\u0000 <annotation>$sigma$</annotation>\u0000 </semantics></math>-compact sets and for closed not strongly dominating sets are <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Π</mi>\u0000 <mn>1</mn>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$operatorname{mathbf {Pi }_1^1}$</annotation>\u0000 </semantics></math>-complete.</p></div>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147569452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convex Structures and Homomorphisms in Ordered Fuzzy Rings: An Algebraic Perspective","authors":"Faisal Mehmood,&nbsp;Fu-Gui Shi,&nbsp;Tahir Mahmood,&nbsp;Muhammad Sajjad,&nbsp;Heng Liu","doi":"10.1002/malq.70009","DOIUrl":"https://doi.org/10.1002/malq.70009","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;This study introduces a novel fuzzy algebraic structure, termed “convex ordered fuzzy subrings” (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;COFSR&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {COFSR}s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;), within the framework of “ordered fuzzy rings” (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;OFR&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {OFR}s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;). We provide a rigorous formalization of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-subrings, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-ideals, and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-convex substructures, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; denotes a complete lattice equipped with a binary, order-compatible t-norm, integrating concepts from convexity theory, fuzzy set theory, and Heyting algebra. We investigate the preservation of convexity and order under ordered fuzzy ring homomorphisms (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;OFRH&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {OFRH}s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) and their kernels, focusing on convex &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-subrings that are closed under fuzzy addition, multiplication, and additive inverses. These substructures induce &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-convex structures with closure properties under Cartesian products and infima, and their behavior under homomorphisms highlights their algebraic and topological significance. Using an axiomatic framework, we establish conditions on lattice-valued binary operations necessary to maintain convexity in fuzzy substructures. Illustrative examples, structural analyses, and interconnections are provided to substantiate these results. This work addresses existing theoretical gaps in fuzzification, ordering, convexity, and homomorphic mappings, offering a unified foundation for the study of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147569531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convex Structures and Homomorphisms in Ordered Fuzzy Rings: An Algebraic Perspective","authors":"Faisal Mehmood,&nbsp;Fu-Gui Shi,&nbsp;Tahir Mahmood,&nbsp;Muhammad Sajjad,&nbsp;Heng Liu","doi":"10.1002/malq.70009","DOIUrl":"10.1002/malq.70009","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;This study introduces a novel fuzzy algebraic structure, termed “convex ordered fuzzy subrings” (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;COFSR&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {COFSR}s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;), within the framework of “ordered fuzzy rings” (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;OFR&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {OFR}s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;). We provide a rigorous formalization of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-subrings, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-ideals, and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-convex substructures, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; denotes a complete lattice equipped with a binary, order-compatible t-norm, integrating concepts from convexity theory, fuzzy set theory, and Heyting algebra. We investigate the preservation of convexity and order under ordered fuzzy ring homomorphisms (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;OFRH&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {OFRH}s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) and their kernels, focusing on convex &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-subrings that are closed under fuzzy addition, multiplication, and additive inverses. These substructures induce &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {L}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-convex structures with closure properties under Cartesian products and infima, and their behavior under homomorphisms highlights their algebraic and topological significance. Using an axiomatic framework, we establish conditions on lattice-valued binary operations necessary to maintain convexity in fuzzy substructures. Illustrative examples, structural analyses, and interconnections are provided to substantiate these results. This work addresses existing theoretical gaps in fuzzification, ordering, convexity, and homomorphic mappings, offering a unified foundation for the study of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"72 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2026-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147569525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}