The classical work of Feferman Vaught gives a powerful, constructive analysis of definability in (generalized) product structures, and certain associated enriched Boolean structures. %structures in terms of definability in the component structures. Here, by closely related methods, but in the special setting of commutative unital rings, we obtain a kind of converse allowing us to determine in interesting cases, when a commutative unital R is elementarily equivalent to a nontrivial product of a family of commutative unital rings R_i. We use this in the model theoretic analysis of residue rings of models of Peano Arithmetic.
费弗曼-沃特的经典著作对（广义）积结构和某些相关的丰富布尔结构的可定义性进行了有力的构造性分析。从成分结构中的可定义性来看%结构。在这里，通过密切相关的方法，但在换元单值环的特殊设置中，我们得到了一种反证，使我们能够在有趣的情况下确定换元单值环 R 何时在元素上等价于换元单值环 R_i 族的非琐积。我们将其用于皮亚诺算术模型的残差环的模型论分析。
{"title":"Commutative unital rings elementarily equivalent to prescribed product rings","authors":"P. D'Aquino, A. Macintyre","doi":"10.4064/fm232-8-2023","DOIUrl":"https://doi.org/10.4064/fm232-8-2023","url":null,"abstract":"The classical work of Feferman Vaught gives a powerful, constructive analysis of definability in (generalized) product structures, and certain associated enriched Boolean structures. %structures in terms of definability in the component structures. Here, by closely related methods, but in the special setting of commutative unital rings, we obtain a kind of converse allowing us to determine in interesting cases, when a commutative unital R is elementarily equivalent to a nontrivial product of a family of commutative unital rings R_i. We use this in the model theoretic analysis of residue rings of models of Peano Arithmetic.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139357538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is shown that Vopv{e}nka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product, $Delta_0$-Separation and Induction along $omega$, then ${rm EST+VP}$ proves the axioms of Infinity, Replacement (thus also Separation) and Powerset. The result was motivated by previous results in cite{Tz14}, as well as by H. Friedman's cite{Fr05}, where a distinction is made among various forms of VP. As a corollary, ${rm EST}+$Foundation$+{rm VP}$=${rm ZF+VP}$, and ${rm EST}+$Foundation$+{rm AC+VP}={rm ZFC+VP}$. Also it is shown that the Foundation axiom is independent from ZF--{Foundation}+${rm VP}$. It is open whether the Axiom of Choice is independent from ${rm ZF+VP}$. A very weak form of choice follows from VP and some similar other forms of choice are introduced.
显示Vopv{e}nka的原理（VP）可以在它的一个弱片段上恢复几乎整个ZF。也就是说，如果EST是由扩展性、空集、配对、并集、笛卡尔乘积、$Delta_0$-分离和归纳的公理沿着$omega$组成的理论，那么$｛rmEST+VP｝$证明了无穷大、替换（因此也是分离）和幂集的公理。这一结果的动机是先前在cite{Tz14}中的结果，以及H.Friedman的cite{Fr05}，其中对各种形式的VP进行了区分。作为必然结果，$｛rm EST｝+$Foundation$+｛rm-VP｝$=$｛lm ZF+VP｝$，以及$｛ rm EST｝+$Foundation$+｝rm-AC+VP｝=｛ rm-ZFC+VP｝$。还证明了Foundation公理独立于ZF—｛Foundation｝+$｛rm-VP｝$。Axiom of Choice是否独立于$｛rm ZF+VP｝$是公开的。VP之后出现了一种非常弱的选择形式，并介绍了一些类似的其他选择形式。
{"title":"Consequences of Vopěnka’s Principle over weak set theories","authors":"A. Tzouvaras","doi":"10.4064/fm982-1-2016","DOIUrl":"https://doi.org/10.4064/fm982-1-2016","url":null,"abstract":"It is shown that Vopv{e}nka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product, $Delta_0$-Separation and Induction along $omega$, then ${rm EST+VP}$ proves the axioms of Infinity, Replacement (thus also Separation) and Powerset. The result was motivated by previous results in cite{Tz14}, as well as by H. Friedman's cite{Fr05}, where a distinction is made among various forms of VP. As a corollary, ${rm EST}+$Foundation$+{rm VP}$=${rm ZF+VP}$, and ${rm EST}+$Foundation$+{rm AC+VP}={rm ZFC+VP}$. Also it is shown that the Foundation axiom is independent from ZF--{Foundation}+${rm VP}$. It is open whether the Axiom of Choice is independent from ${rm ZF+VP}$. A very weak form of choice follows from VP and some similar other forms of choice are introduced.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45923563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A bounded sequence of bitransitive and capture Sierpiński curve Julia sets for 3-circle inversions","authors":"Daniel M. Look","doi":"10.4064/fm166-4-2023","DOIUrl":"https://doi.org/10.4064/fm166-4-2023","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70215788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that if $G$ is a topological graph, and $fcolon Gto G$ is a continuous map, then the induced map $tilde {f}$ defined on the hyperspace $C(G)$ of all connected subsets of $G$ by the natural formula $tilde {f}(C)=f(C)$ carries the same entro
{"title":"On recurrence and entropy in the hyperspace of continua in dimension one","authors":"Domagoj Jelić, Piotr Oprocha","doi":"10.4064/fm235-4-2023","DOIUrl":"https://doi.org/10.4064/fm235-4-2023","url":null,"abstract":"We show that if $G$ is a topological graph, and $fcolon Gto G$ is a continuous map, then the induced map $tilde {f}$ defined on the hyperspace $C(G)$ of all connected subsets of $G$ by the natural formula $tilde {f}(C)=f(C)$ carries the same entro","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135056928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the context of holomorphic families of endomorphisms of $mathbb P^k$, we prove that stability in the sense of [F. Berteloot et al., Ann. Sci. École Norm. Sup. 51 (2018)] is equivalent to a summability condition for the post-critical mass and to the co
{"title":"Ramification current, post-critical normality and stability of holomorphic endomorphisms of $mathbb P^k$","authors":"François Berteloot, Maxence Brévard","doi":"10.4064/fm279-6-2023","DOIUrl":"https://doi.org/10.4064/fm279-6-2023","url":null,"abstract":"In the context of holomorphic families of endomorphisms of $mathbb P^k$, we prove that stability in the sense of [F. Berteloot et al., Ann. Sci. École Norm. Sup. 51 (2018)] is equivalent to a summability condition for the post-critical mass and to the co","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135957904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dimension of images and graphs of little Lipschitz functions","authors":"","doi":"10.4064/fm147-12-2022","DOIUrl":"https://doi.org/10.4064/fm147-12-2022","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70203168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We are dealing with the complexity of the homeomorphism relation on some classes of metrizable compacta from the viewpoint of invariant descriptive set theory. We prove that the homeomorphism relation for absolute retracts in the plane is Borel bireducibl
{"title":"The complexity of homeomorphism relations on some classes of compacta with bounded topological dimension","authors":"Jan Dudák, Benjamin Vejnar","doi":"10.4064/fm164-8-2023","DOIUrl":"https://doi.org/10.4064/fm164-8-2023","url":null,"abstract":"We are dealing with the complexity of the homeomorphism relation on some classes of metrizable compacta from the viewpoint of invariant descriptive set theory. We prove that the homeomorphism relation for absolute retracts in the plane is Borel bireducibl","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New axioms for the lattice-ordered groups existentially closed in ${bf W}^{+}$","authors":"Brian Wynne","doi":"10.4064/fm280-6-2023","DOIUrl":"https://doi.org/10.4064/fm280-6-2023","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70292610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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