{"id":350033,"date":"2017-01-10T10:44:59","date_gmt":"2017-01-10T18:44:59","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=350033"},"modified":"2018-10-16T22:02:53","modified_gmt":"2018-10-17T05:02:53","slug":"113-finite-rigid-structures","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/113-finite-rigid-structures\/","title":{"rendered":"On Finite Rigid Structures"},"content":{"rendered":"<p>This is related to the problem of defining linear order on finite structures. If a linear order is definable on a finite structure <i>A<\/i> then <i>A<\/i> is rigid (which means that its only automorphism is the identity). There had been a suspicion that if <i>K<\/i> is the collection of all finite structures of a finitely axiomatizable class and if every <i>K<\/i> structure is rigid, then <i>K<\/i> permits a relatively simple uniform definition of linear order. That happens to be not the case. The main result of the paper is a probabilistic construction of finite rigid graphs. Using that construction, we exhibit a finitely axiomatizable class of finite rigid structures (called multipedes) such that no L<sup>\u03c9<\/sup><sub>\u221e, \u03c9<\/sub> sentence \u03c6 with counting quantifiers defines a linear order in all the structures. Furthermore, \u03c6 does not distinguish between a sufficiently large multipede <i>M<\/i> and a multipede <i>M&#8217;<\/i> obtained from <i>M<\/i> by moving a &#8220;shoe&#8221; to another foot of the same segment.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is related to the problem of defining linear order on finite structures. If a linear order is definable on a finite structure A then A is rigid (which means that its only automorphism is the identity). There had been a suspicion that if K is the collection of all finite structures of a finitely [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"Journal of Symbolic Logic","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"2","msr_journal":"Journal of Symbolic 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