Neeraj K. Dhanwani, Hitesh Raundal, Mahender Singh
Unifying various constructions of quandles including Coxeter quandles, free quandles, knot quandles of prime knots and Dehn quandles of orientable surfaces, we introduce Dehn quandles of groups with respect to their subsets. We prove that the enveloping g
{"title":"Dehn quandles of groups and orientable surfaces","authors":"Neeraj K. Dhanwani, Hitesh Raundal, Mahender Singh","doi":"10.4064/fm271-6-2023","DOIUrl":"https://doi.org/10.4064/fm271-6-2023","url":null,"abstract":"Unifying various constructions of quandles including Coxeter quandles, free quandles, knot quandles of prime knots and Dehn quandles of orientable surfaces, we introduce Dehn quandles of groups with respect to their subsets. We prove that the enveloping g","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136297312","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 this paper we study the behaviour of selective separability properties in the class of Frech'{e}t-Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Frech'{e}t-Urysohn (hence $R$-separable and selectively separable) space which is not $H$-separable; assuming $mathfrak{p}=mathfrak{c}$, we construct such an example which is also zero-dimensional and $alpha_{4}$. Also, motivated by a result of Barman and Dow stating that the product of two countable Frech'{e}t-Urysohn spaces is $M$-separable under PFA, we show that the MA is not sufficient here. In the last section we prove that in the Laver model, the product of any two $H$-separable spaces is $mH$-separable.
{"title":"Selective separability properties of Fréchet–Urysohn spaces and their products","authors":"Serhii Bardyla, Fortunato Maesano, Lyubomyr Zdomskyy","doi":"10.4064/fm230522-13-10","DOIUrl":"https://doi.org/10.4064/fm230522-13-10","url":null,"abstract":"In this paper we study the behaviour of selective separability properties in the class of Frech'{e}t-Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Frech'{e}t-Urysohn (hence $R$-separable and selectively separable) space which is not $H$-separable; assuming $mathfrak{p}=mathfrak{c}$, we construct such an example which is also zero-dimensional and $alpha_{4}$. Also, motivated by a result of Barman and Dow stating that the product of two countable Frech'{e}t-Urysohn spaces is $M$-separable under PFA, we show that the MA is not sufficient here. In the last section we prove that in the Laver model, the product of any two $H$-separable spaces is $mH$-separable.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135759595","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 for any two elementary equivalent structures $mathbf M, mathbf N$ of size at most continuum in a countable language, $mathbf M^{omega }/ mathcal U simeq mathbf N^omega / mathcal U$ for some ultrafilter $mathcal U$ on $omega ,$ t
我们证明了如果对于任意两个基本等效结构$mathbf M, mathbf N$在可数语言中最多连续体的大小，$mathbf M^{omega }/ mathcal U simeq mathbf N^omega / mathcal U$对于某些超过滤器$mathcal U$在$omega ,$ t上
{"title":"The Keisler–Shelah isomorphism theorem and the continuum hypothesis","authors":"Mohammad Golshani, Saharon Shelah","doi":"10.4064/fm198-5-2022","DOIUrl":"https://doi.org/10.4064/fm198-5-2022","url":null,"abstract":"We show that if for any two elementary equivalent structures $mathbf M, mathbf N$ of size at most continuum in a countable language, $mathbf M^{omega }/ mathcal U simeq mathbf N^omega / mathcal U$ for some ultrafilter $mathcal U$ on $omega ,$ t","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181016","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}
The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter $t$ while producing categories of representations of the symmetric group when modded out by the ideal
{"title":"Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category","authors":"Mikhail Khovanov, Radmila Sazdanovic","doi":"10.4064/fm283-8-2023","DOIUrl":"https://doi.org/10.4064/fm283-8-2023","url":null,"abstract":"The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter $t$ while producing categories of representations of the symmetric group when modded out by the ideal","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"260 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136092727","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":"Almost everywhere convergence of nets of operators and weak type maximal inequalities","authors":"","doi":"10.4064/fm272-2-2023","DOIUrl":"https://doi.org/10.4064/fm272-2-2023","url":null,"abstract":"","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70292500","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}
A flow is a directed space structure on a homotopy type. It is already known that the underlying homotopy type of the realization of a precubical set as a flow is homotopy equivalent to the realization of the precubical set as a topological space. This realization depends on the non-canonical choice of a q-cofibrant replacement. We construct a new realization functor from precubical sets to flows which is homotopy equivalent to the previous one and which does not depend on the choice of any cofibrant replacement functor. The main tool is the notion of natural $d$-path introduced by Raussen. The flow we obtain for a given precubical set is not anymore q-cofibrant but is still m-cofibrant. As an application, we prove that the space of execution paths of the realization of a precubical set as a flow is homotopy equivalent to the space of nonconstant $d$-paths between vertices in the geometric realization of the precubical set.
{"title":"Comparing cubical and globular directed paths","authors":"P. Gaucher","doi":"10.4064/fm219-3-2023","DOIUrl":"https://doi.org/10.4064/fm219-3-2023","url":null,"abstract":"A flow is a directed space structure on a homotopy type. It is already known that the underlying homotopy type of the realization of a precubical set as a flow is homotopy equivalent to the realization of the precubical set as a topological space. This realization depends on the non-canonical choice of a q-cofibrant replacement. We construct a new realization functor from precubical sets to flows which is homotopy equivalent to the previous one and which does not depend on the choice of any cofibrant replacement functor. The main tool is the notion of natural $d$-path introduced by Raussen. The flow we obtain for a given precubical set is not anymore q-cofibrant but is still m-cofibrant. As an application, we prove that the space of execution paths of the realization of a precubical set as a flow is homotopy equivalent to the space of nonconstant $d$-paths between vertices in the geometric realization of the precubical set.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41645554","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}
Let $E$ be a countable Borel equivalence relation on the space $mathcal{E}_{infty}$ of all infinite partitions of the natural numbers. We show that $E$ coincides with equality below a Carlson-Simpson generic element of $mathcal{E}_{infty}$. In contrast, we show that there is a hypersmooth equivalence relation on $mathcal{E}_{infty}$ which is Borel bireducible with $E_1$ on every Carlson-Simpson cube. Our arguments are classical and require no background in forcing.
{"title":"Every CBER is smooth below\u0000the Carlson–Simpson generic partition","authors":"Aristotelis Panagiotopoulos, Allison Wang","doi":"10.4064/fm255-12-2022","DOIUrl":"https://doi.org/10.4064/fm255-12-2022","url":null,"abstract":"Let $E$ be a countable Borel equivalence relation on the space $mathcal{E}_{infty}$ of all infinite partitions of the natural numbers. We show that $E$ coincides with equality below a Carlson-Simpson generic element of $mathcal{E}_{infty}$. In contrast, we show that there is a hypersmooth equivalence relation on $mathcal{E}_{infty}$ which is Borel bireducible with $E_1$ on every Carlson-Simpson cube. Our arguments are classical and require no background in forcing.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42308945","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}
. All spaces below are T 0 and crowded (i.e. have no isolated points). For n ≤ ω let M ( n ) be the statement that there are n measurable cardinals and Π( n ) ( Π + ( n ) ) that there are n +1 (0-dimensional T 2 ) spaces whose product is irresolvable. We prove that M (1) , Π(1) and Π + (1) are equiconsistent. For 1 < n < ω we show that CON ( M ( n )) implies CON (Π + ( n )) . Finally, CON ( M ( ω )) implies the consistency of having infinitely many crowded 0-dimensional T 2 -spaces such that the product of any finitely many of them is irresolvable. These settle old problems of Malychin from [11]. Concerning an even older question of Ceder and Pearson in [1], we show that the following are consistent modulo a measurable cardinal: These significantly improve Eckertson
{"title":"On resolvability of products","authors":"I. Juh'asz, L. Soukup, Z. Szentmikl'ossy","doi":"10.4064/fm244-10-2022","DOIUrl":"https://doi.org/10.4064/fm244-10-2022","url":null,"abstract":". All spaces below are T 0 and crowded (i.e. have no isolated points). For n ≤ ω let M ( n ) be the statement that there are n measurable cardinals and Π( n ) ( Π + ( n ) ) that there are n +1 (0-dimensional T 2 ) spaces whose product is irresolvable. We prove that M (1) , Π(1) and Π + (1) are equiconsistent. For 1 < n < ω we show that CON ( M ( n )) implies CON (Π + ( n )) . Finally, CON ( M ( ω )) implies the consistency of having infinitely many crowded 0-dimensional T 2 -spaces such that the product of any finitely many of them is irresolvable. These settle old problems of Malychin from [11]. Concerning an even older question of Ceder and Pearson in [1], we show that the following are consistent modulo a measurable cardinal: These significantly improve Eckertson","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44113482","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 say that a finitely additive probability measure $mu$ on $omega$ is emph{a P-measure} if it vanishes on points and for each decreasing sequence $(E_n)$ of infinite subsets of $omega$ there is $Esubseteqomega$ such that $Esubseteq^* E_n$ for each $ninomega$ and $mu(E) = lim_{ntoinfty}mu(E_n)$. Thus, P-measures generalize in a natural way P-points and it is known that, similarly as in the case of P-points, their existence is independent of $mathsf{ZFC}$. In this paper we show that there is a P-measure in the model obtained by adding any number of random reals to a model of $mathsf{CH}$. As a corollary, we obtain that in the classical random model $omega^*$ contains a nowhere dense ccc closed P-set.
{"title":"There is a P-measure in the random model","authors":"Piotr Borodulin-Nadzieja, Damian Sobota","doi":"10.4064/fm277-3-2023","DOIUrl":"https://doi.org/10.4064/fm277-3-2023","url":null,"abstract":"We say that a finitely additive probability measure $mu$ on $omega$ is emph{a P-measure} if it vanishes on points and for each decreasing sequence $(E_n)$ of infinite subsets of $omega$ there is $Esubseteqomega$ such that $Esubseteq^* E_n$ for each $ninomega$ and $mu(E) = lim_{ntoinfty}mu(E_n)$. Thus, P-measures generalize in a natural way P-points and it is known that, similarly as in the case of P-points, their existence is independent of $mathsf{ZFC}$. In this paper we show that there is a P-measure in the model obtained by adding any number of random reals to a model of $mathsf{CH}$. As a corollary, we obtain that in the classical random model $omega^*$ contains a nowhere dense ccc closed P-set.","PeriodicalId":55138,"journal":{"name":"Fundamenta Mathematicae","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42339551","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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