{"title":"序理论树的诱导间性","authors":"B. Courcelle","doi":"10.46298/dmtcs.7288","DOIUrl":null,"url":null,"abstract":"The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x<y<z∨z<y<x, and the corresponding betweenness structure is (N,B). The class of betweenness structures of linear orders is first-order definable. That of partial orders is monadic second-order definable. An order-theoretic tree is a partial order such that the set of elements larger that any element is linearly ordered and any two elements have an upper-bound. Finite or infinite rooted trees ordered by the ancestor relation are order-theoretic trees. In an order-theoretic tree, B(x,y,z) means that x<y<z or z<y<x or x<y≤x⊔z or z<y≤x⊔z, where x⊔z is the least upper-bound of incomparable elements x and z. In a previous article, we established that the corresponding class of betweenness structures is monadic second-order definable.We prove here that the induced substructures of the betweenness structures of the countable order-theoretic trees form a monadic second-order definable class, denoted by IBO. The proof uses a variant of cographs, the partitioned probe cographs, and their known six finite minimal excluded induced subgraphs called the bounds of the class. This proof links two apparently unrelated topics: cographs and order-theoretic trees.However, the class IBO has finitely many bounds, i.e., minimal excluded finite induced substructures. Hence it is first-order definable. The proof of finiteness uses well-quasi-orders and does not provide the finite list of bounds. Hence, the associated first-order defining sentence is not known.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"43 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Induced betweenness in order-theoretic trees\",\"authors\":\"B. Courcelle\",\"doi\":\"10.46298/dmtcs.7288\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x<y<z∨z<y<x, and the corresponding betweenness structure is (N,B). The class of betweenness structures of linear orders is first-order definable. That of partial orders is monadic second-order definable. An order-theoretic tree is a partial order such that the set of elements larger that any element is linearly ordered and any two elements have an upper-bound. Finite or infinite rooted trees ordered by the ancestor relation are order-theoretic trees. In an order-theoretic tree, B(x,y,z) means that x<y<z or z<y<x or x<y≤x⊔z or z<y≤x⊔z, where x⊔z is the least upper-bound of incomparable elements x and z. In a previous article, we established that the corresponding class of betweenness structures is monadic second-order definable.We prove here that the induced substructures of the betweenness structures of the countable order-theoretic trees form a monadic second-order definable class, denoted by IBO. The proof uses a variant of cographs, the partitioned probe cographs, and their known six finite minimal excluded induced subgraphs called the bounds of the class. This proof links two apparently unrelated topics: cographs and order-theoretic trees.However, the class IBO has finitely many bounds, i.e., minimal excluded finite induced substructures. Hence it is first-order definable. The proof of finiteness uses well-quasi-orders and does not provide the finite list of bounds. Hence, the associated first-order defining sentence is not known.\",\"PeriodicalId\":110830,\"journal\":{\"name\":\"Discret. Math. Theor. Comput. Sci.\",\"volume\":\"43 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-11-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discret. Math. Theor. Comput. Sci.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.46298/dmtcs.7288\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discret. Math. Theor. Comput. Sci.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.46298/dmtcs.7288","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x
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