{"id":119,"date":"2026-08-15T09:45:03","date_gmt":"2026-08-15T09:45:03","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=119"},"modified":"2026-09-19T17:19:34","modified_gmt":"2026-09-19T17:19:34","slug":"does-widening-a-definition-weaken-it","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/15\/does-widening-a-definition-weaken-it\/","title":{"rendered":"Does widening a definition weaken it?"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Why do we need another definition?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We have the definition for generator being an element of a group.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But a sturdy element is instead an element of a set.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We could widen the definition for generator to be an element of a set.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Would that create problems?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Should we say super generator instead of sturdy element?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But this might require or imply a generator could be an element of a set.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There is nothing wrong with describing the results of using a sturdy element modulo enclosing set as being a cyclic group.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">However to call those results a cyclic subgroup implies a parent group and that is not always the case.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We could widen the definition of subgroup to include a parent set also.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Would that create problems?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A new definition &#8220;super element&#8221; avoids having to widen existing definitions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Perhaps because a debate about widening generator or subgroup has not yet happened shows that these elements and their behaviour have not yet been studied more widely.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#8220;They are just generators&#8221;<br>But they do not belong to a group [in all cases]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#8220;They are super generators&#8221;<br>We need to be careful we do not imply group membership [in all cases] by reusing the term generator<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#8220;They generate cyclic subgroups&#8221;<br>We need to be careful we do not imply group membership [in all cases] by the current definition of a subgroup<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But wouldn&#8217;t having a sturdy element require a new notation?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If we reuse the angle brackets that we <a href=\"https:\/\/en.wikipedia.org\/wiki\/Generating_set_of_a_group#Finitely_generated_group\">use for generator<\/a> typically then we could risk confusion.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">How about double angle brackets <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&lt;&lt;-1+2^t>> for the sturdy element -1+2^t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&lt;&lt;1+2^t>> for the sturdy element 1+2^t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&lt;&lt;-1+3^t>> for the sturdy element -1+3^t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&lt;&lt;1+3^t>> for the sturdy element 1+3^t<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Each of those sturdy elements has a context set or enclosing set so should be quoted in more complete form as shown next.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"608\" height=\"78\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/superelement1plus2toT.png\" alt=\"\" class=\"wp-image-138\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/superelement1plus2toT.png 608w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/08\/superelement1plus2toT-300x38.png 300w\" sizes=\"auto, (max-width: 608px) 100vw, 608px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Why do we need another definition? We have the definition for generator being an element of a group. But a sturdy element is instead an element of a set. We could widen the definition for generator to be an element of a set. Would that create problems? Should we say super generator instead of sturdy [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,6],"tags":[11,8],"class_list":["post-119","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-numbertheory","tag-definition","tag-group"],"_links":{"self":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/119","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/comments?post=119"}],"version-history":[{"count":6,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/119\/revisions"}],"predecessor-version":[{"id":340,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/119\/revisions\/340"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=119"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=119"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=119"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}