Pub Date : 2019-10-08DOI: 10.4467/20842589rm.19.003.10651
Boris Šobot
The paper first covers several properties of the extension of the divisibility relation to a set *N of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the Stone–Cech compactification βN, proving that the divisibility of ultrafilters introduced by the author is equivalent to divisibility of some elements belonging to their respective monads in an enlargement. Some earlier results on ultrafilters on lower levels on the divisibility hierarchy are illuminated by nonstandard methods. Using limits by ultrafilters we obtain results on ultrafilters above these finite levels, showing that for them a distribution by levels is not possible. Received 16 July 2018 AMS subject classification: Primary 54D80; Secondary 11U10, 03H15, 54D35 Keywords: divisibility, nonstandard integer, Stone-Cech compactification, ultrafilter
{"title":"Divisibility in beta N and *N","authors":"Boris Šobot","doi":"10.4467/20842589rm.19.003.10651","DOIUrl":"https://doi.org/10.4467/20842589rm.19.003.10651","url":null,"abstract":"The paper first covers several properties of the extension of the divisibility relation to a set *N of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the Stone–Cech compactification βN, proving that the divisibility of ultrafilters introduced by the author is equivalent to divisibility of some elements belonging to their respective monads in an enlargement. Some earlier results on ultrafilters on lower levels on the divisibility hierarchy are illuminated by nonstandard methods. Using limits by ultrafilters we obtain results on ultrafilters above these finite levels, showing that for them a distribution by levels is not possible. \u0000Received 16 July 2018 \u0000AMS subject classification: Primary 54D80; Secondary 11U10, 03H15, 54D35 \u0000 \u0000Keywords: divisibility, nonstandard integer, Stone-Cech compactification, ultrafilter","PeriodicalId":48992,"journal":{"name":"Reports on Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49645883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4467/20842589rm.19.002.10650
R. Camerlo
It is proved that the Tang-Pequignot reducibility (or reducibility by relatively continuous relations) on a second countable, T0 space X either coincides with the Wadge reducibility for the given topology, or there is no topology on X that can turn it into Wadge reducibility.
{"title":"Continuous reducibility: functions versus relations","authors":"R. Camerlo","doi":"10.4467/20842589rm.19.002.10650","DOIUrl":"https://doi.org/10.4467/20842589rm.19.002.10650","url":null,"abstract":"It is proved that the Tang-Pequignot reducibility (or reducibility by relatively continuous relations) on a second countable, T0 space X either coincides with the Wadge reducibility for the given topology, or there is no topology on X that can turn it into Wadge reducibility.","PeriodicalId":48992,"journal":{"name":"Reports on Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70985993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4467/20842589rm.19.007.10655
Éva Jungábel
A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. A point-line geometry is a non-empty set of elements called points, together with a collection of subsets, called lines, in a way that every line contains at least two points and any pair of points is contained in at most one line. A line which contains more than two points is called a regular line. Point-line geometries can alternatively be formalised as relational structures. We establish a correspondence between the point-line geometries investigated in this paper and the firstorder structures with a single ternary relation L satisfying certain axioms (i.e. that the class of point-line geometries corresponds to a subclass of 3-uniform hypergraphs). We characterise the homomorphism-homogeneous point-line geometries with two regular non-intersecting lines. Homomorphism-homogeneous pointline geometries containing two regular intersecting lines have already been classified by Masulovic.
{"title":"On homomorphism-homogeneous point-line geometries","authors":"Éva Jungábel","doi":"10.4467/20842589rm.19.007.10655","DOIUrl":"https://doi.org/10.4467/20842589rm.19.007.10655","url":null,"abstract":"A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. A point-line geometry is a non-empty set of elements called points, together with a collection of subsets, called lines, in a way that every line contains at least two points and any pair of points is contained in at most one line. A line which contains more than two points is called a regular line. Point-line geometries can alternatively be formalised as relational structures. We establish a correspondence between the point-line geometries investigated in this paper and the firstorder structures with a single ternary relation L satisfying certain axioms (i.e. that the class of point-line geometries corresponds to a subclass of 3-uniform hypergraphs). We characterise the homomorphism-homogeneous point-line geometries with two regular non-intersecting lines. Homomorphism-homogeneous pointline geometries containing two regular intersecting lines have already been classified by Masulovic.","PeriodicalId":48992,"journal":{"name":"Reports on Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70986010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-09-14DOI: 10.4467/20842589RM.16.009.5286
Hitoshi Omori
A b s t r a c t. The present note corrects an error made by the author in answering an open problem of axiomatizing an expansion of Nelson’s logic introduced by Heinrich Wansing. It also gives a correct axiomatization that answers the problem by importing some results on subintuitionistic logics presented by Greg Restall.
{"title":"A Note on Wansing's expansion of Nelson's logic","authors":"Hitoshi Omori","doi":"10.4467/20842589RM.16.009.5286","DOIUrl":"https://doi.org/10.4467/20842589RM.16.009.5286","url":null,"abstract":"A b s t r a c t. The present note corrects an error made by the author in answering an open problem of axiomatizing an expansion of Nelson’s logic introduced by Heinrich Wansing. It also gives a correct axiomatization that answers the problem by importing some results on subintuitionistic logics presented by Greg Restall.","PeriodicalId":48992,"journal":{"name":"Reports on Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70986330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-12-15DOI: 10.4467/20842589RM.11.008.0286
Yasusi Hasimoto, Akio Maruyama
We describe properties of simply axiomatized modal logics, which are called pseudo-Euclidean modal logics. We will then give a complete description of the inclusion relationship among these logics by showing inclusion relationships for pairs of their logics with fixed m and n.
{"title":"Inclusions Between Pseudo-euclidean Modal Logics","authors":"Yasusi Hasimoto, Akio Maruyama","doi":"10.4467/20842589RM.11.008.0286","DOIUrl":"https://doi.org/10.4467/20842589RM.11.008.0286","url":null,"abstract":"We describe properties of simply axiomatized modal logics, which are called pseudo-Euclidean modal logics. We will then give a complete description of the inclusion relationship among these logics by showing inclusion relationships for pairs of their logics with fixed m and n.","PeriodicalId":48992,"journal":{"name":"Reports on Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2011-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70986252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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