{"title":"Antichains of copies of ultrahomogeneous structures","authors":"Miloš S. Kurilić, Boriša Kuzeljević","doi":"10.1007/s00153-022-00817-7","DOIUrl":null,"url":null,"abstract":"<div><p>We investigate possible cardinalities of maximal antichains in the poset of copies <span>\\(\\langle {\\mathbb {P}}(\\mathbb X),\\subseteq \\rangle \\)</span> of a countable ultrahomogeneous relational structure <span>\\({{\\mathbb {X}}}\\)</span>. It turns out that if the age of <span>\\({{\\mathbb {X}}}\\)</span> has the strong amalgamation property, then, defining a copy of <span>\\({{\\mathbb {X}}}\\)</span> to be large iff it has infinite intersection with each orbit of <span>\\({{\\mathbb {X}}}\\)</span>, the structure <span>\\({{\\mathbb {X}}}\\)</span> can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (maximal) antichains of size continuum in the poset <span>\\({{\\mathbb {P}}}({{\\mathbb {X}}})\\)</span>. Finally, we show that the posets of copies of all countable ultrahomogeneous partial orders contain maximal antichains of cardinality continuum and determine which of them contain countable maximal antichains. That holds, in particular, for the generic (universal ultrahomogeneous) poset.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-022-00817-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Arts and Humanities","Score":null,"Total":0}
引用次数: 2
Abstract
We investigate possible cardinalities of maximal antichains in the poset of copies \(\langle {\mathbb {P}}(\mathbb X),\subseteq \rangle \) of a countable ultrahomogeneous relational structure \({{\mathbb {X}}}\). It turns out that if the age of \({{\mathbb {X}}}\) has the strong amalgamation property, then, defining a copy of \({{\mathbb {X}}}\) to be large iff it has infinite intersection with each orbit of \({{\mathbb {X}}}\), the structure \({{\mathbb {X}}}\) can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (maximal) antichains of size continuum in the poset \({{\mathbb {P}}}({{\mathbb {X}}})\). Finally, we show that the posets of copies of all countable ultrahomogeneous partial orders contain maximal antichains of cardinality continuum and determine which of them contain countable maximal antichains. That holds, in particular, for the generic (universal ultrahomogeneous) poset.
期刊介绍:
The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.