[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-157779-105":59,"doc-detail-157779-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","the-strongest-t-norm-for-fuzzy-metric-spaces","THE STRONGEST T-NORM FOR FUZZY METRIC SPACES","","For any given positive continuous t-norm, the paper constructs a fuzzy metric space in the sense of George and Veeramani such that the chosen t-norm is the strongest one compatible with that space. It also analyzes the dual question for a fixed George–Veeramani fuzzy metric space, showing that no strongest continuous t-norm exists in general. As an application, the work establishes the existence of infinitely many non-isometric fuzzy metrics on an infinite set.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/the-strongest-t-norm-for-fuzzy-metric-spaces/157779/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/the-strongest-t-norm-for-fuzzy-metric-spaces/157779.png","ImageObject",300,407,{"name":92,"@type":93},"\tJames","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-01","2026-08-29",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How does the paper address the problem of finding a fuzzy metric space for a given t-norm?","Question",{"text":113,"@type":114},"It proves that for any given positive continuous t-norm, there exists a George–Veeramani fuzzy metric space in which that t-norm is the strongest.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What is the conclusion for the opposite question: does a George–Veeramani fuzzy metric space always have a strongest continuous t-norm?",{"text":118,"@type":114},"The paper shows the answer is negative for continuous t-norms, meaning a strongest one does not necessarily exist.",{"name":120,"@type":111,"acceptedAnswer":121},"What application is derived from the main results?",{"text":122,"@type":114},"It demonstrates that on an infinite set there exist infinitely many non-isometric George–Veeramani fuzzy metrics.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},157779,1787988642,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":46,"preview_page_count":4},2336474466412,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","THE STRONGEST T-NORM FOR FUZZY METRIC SPACES  \nDong Qiu, Weiquan Zhang  \nIn this paper, we prove that for a given positive continuous t-norm there is a fuzzy metric space in the sense of George and Veeramani, for which the given t-norm is the strongest one. For the opposite problem, we obtain that there is a fuzzy metric space for which there is no strongest t-norm. As an application of the main results, it is shown that there are in􀀌nite non-isometric fuzzy metrics on an in􀀌nite set.  \nKeywords: fuzzy metric space, t-norm, isometry, analysis Classi􀀌cation: 93E12, 62A10  \n1. INTRODUCTION  \nTriangular norms (t-norms for short), introduced by Schweizer and Sklar [9], play a key role in the theory of fuzzy metric spaces. For a given fuzzy metric space in the sense of George and Veeramani [2](GV fuzzy metric space for short), there is generally more than one t-norm to choose from. Speci􀀌cally, if (X; M; T ) is a GV fuzzy metric space and a continuous t-norm T0 is weaker than T , then (X; M; T0 ) is also a GV fuzzy metric space [8] . Furthermore, the properties of the triangle inequality are greatly dependent on the corresponding t-norm in the GV fuzzy metric space. It follows, in general, that the stronger t-norm, the more information one has about the structure of the GV fuzzy metric space. Hence it is natural to ask whether, for a given t-norm, there is a corresponding GV fuzzy metric space for which the given t-norm is strongest. Similarly it is interesting to know whether, for a given GV fuzzy metric space, there is a strongest t-norm.  \nIt should be noted that the same issues have been discussed for probabilistic metric spaces in [11] . Since any fuzzy metric space in the sense of Kramosil and Mich􀀓alek [6]  \n(KM fuzzy metric space for short) is equivalent to a Menger space [6], the results of [11] hold for the KM fuzzy metric spaces. However, because a GV fuzzy metric space has the stricter \\separation\" condition and continuity condition than a KM fuzzy metric space, the conclusions and the constructive proofs in [11] no longer apply to the GV fuzzy metric spaces. In this paper, we solve the former problem for all of the positive continuous t-norms by a new constructive way which is di􀀋erent from the one in [11] . We also investigate the latter problem. Unfortunately, the answer is negative for continuous t-norms. Finally, as an application of the main results, it is shown that there are in􀀌nite non-isometric GV fuzzy metrics on an in􀀌nite set.  \n142 D. QIU, W. ZHANG  \n2. PRELIMINARIES  \nThroughout this paper the letters N, Q and R will denote the set of all positive integers, rational numbers and real numbers, respectively.  \nDe􀀌nition 2.1 . (Klement et al. [5]) A t-norm is a binary operation on the unit interval [0 ; 1], i. e. , a function T : [0 ; 1]2 ! [0 ; 1], such that for all a; b; c; d 2 [0 ; 1] the following four axioms are satis􀀌ed:  \n(T-1) T (a; 1) = a. (boundary condition)  \n(T-2) T (a; b) 􀀔 T(c; d) whenever a 􀀔 c and b 􀀔 d. (monotonicity)  \n(T-3) T (a; b) = T (b; a) . (commutativity)(T-4) T (a; T(b; c)) = T (T (a; b); c) . (associativity) A t-norm T is said to be continuous if it is a continuous function on [0 ; 1]2 .  \nExample 2.1 . (Klement et al. [5]) The following are the four basic t-norms TM , TP , TL and TD given by, respectively, TM (x; y) = minfx; yg; TP (x; y) = x 􀀁 y; TL (x; y) = maxfx + y 􀀀 1 ; 0g and  \nTD (x; y) = 􀀚 0minfx; yg ifoth(xer; )is2e . [0 ; 1[2 ;  \nThe t-norms TM , TP and TL are continuous but TD is not.  \nDe􀀌nition 2.2 . (Klement et al. [5]) If, for two t-norms T1 and T2 , T1 (a; b) 􀀔 T2 (a; b) holds for all (a; b) 2 [0 ; 1]2 , then we say that T2 is stronger than T1 , and we write in this case T1 􀀔 T2 . We write T1 \u003C T2 whenever T1 􀀔 T2 and T1  T2 , and we say that T2 is strictly stronger than T1 .  \nAlthough it is true that TD \u003C TL \u003C TP \u003C TM and, for an arbitrary t-norm T , TD 􀀔 T 􀀔 TM , the relation 􀀔 is not a total order, i. e., there are incomparable t-norms.  \nExample 2.2 . The fami","cbCaiumTwZzb0nzp","https://ap.wps.com/l/cbCaiumTwZzb0nzp","pdf",170483,"English","# Introduction\n# Preliminaries\n## t-norms and continuity\n## Comparison of t-norm strength\n## Related metric-space notions\n# Main results and application","[{\"question\":\"How does the paper address the problem of finding a fuzzy metric space for a given t-norm?\",\"answer\":\"It proves that for any given positive continuous t-norm, there exists a George–Veeramani fuzzy metric space in which that t-norm is the strongest.\"},{\"question\":\"What is the conclusion for the opposite question: does a George–Veeramani fuzzy metric space always have a strongest continuous t-norm?\",\"answer\":\"The paper shows the answer is negative for continuous t-norms, meaning a strongest one does not necessarily exist.\"},{\"question\":\"What application is derived from the main results?\",\"answer\":\"It demonstrates that on an infinite set there exist infinitely many non-isometric George–Veeramani fuzzy metrics.\"}]","THE STRONGEST T-NORM FOR FUZZY METRIC SPACES | PDF"]