Pub Date : 2024-01-05DOI: 10.1016/j.apal.2023.103403
Liling Ko
Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees , it is not known how one can characterize the degrees below which L can be embedded. Two important characterizations are of the and lattices, where the lattices are embedded below d if and only if d contains sets of “fickleness” >ω and respectively. We work towards finding a lattice that characterizes the levels above , the first non-trivial level after ω. We considered lattices that are as “short” in height and “narrow” in width as and , but the lattices characterize also the >ω or levels, if the lattices are not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some previously considered lattices, but the removals did not change the levels characterized. We discovered three lattices besides that also characterize the -levels. Our search for -candidates can therefore be reduced to the lattice-theoretic problem of finding lattices that do not contain any of the four -lattices as sublattices.
{"title":"Towards characterizing the >ω2-fickle recursively enumerable Turing degrees","authors":"Liling Ko","doi":"10.1016/j.apal.2023.103403","DOIUrl":"10.1016/j.apal.2023.103403","url":null,"abstract":"<div><p><span>Given a finite lattice </span><em>L</em><span> that can be embedded in the recursively enumerable (r.e.) Turing degrees </span><span><math><mo>〈</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo><msub><mrow><mo>≤</mo></mrow><mrow><mi>T</mi></mrow></msub><mo>〉</mo></math></span>, it is not known how one can characterize the degrees <span><math><mi>d</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>T</mi></mrow></msub></math></span> below which <em>L</em> can be embedded. Two important characterizations are of the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>7</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> lattices, where the lattices are embedded below <strong>d</strong> if and only if <strong>d</strong> contains sets of “<em>fickleness</em>” ><em>ω</em> and <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> respectively. We work towards finding a lattice that characterizes the levels above <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, the first non-trivial level after <em>ω</em>. We considered lattices that are as “short” in height and “narrow” in width as <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>7</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, but the lattices characterize also the ><em>ω</em> or <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> levels, if the lattices are not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some previously considered lattices, but the removals did not change the levels characterized. We discovered three lattices besides <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> that also characterize the <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>-levels. Our search for <span><math><mo>></mo><msup><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-candidates can therefore be reduced to the lattice-theoretic problem of finding lattices that do not contain any of the four <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span><span>-lattices as sublattices.</span></p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139104768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-16DOI: 10.1016/j.apal.2023.103391
Paolo Aglianò , Sara Ugolini
In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of -algebras, i.e., bounded commutative integral residuated lattices generated by chains.
{"title":"Structural and universal completeness in algebra and logic","authors":"Paolo Aglianò , Sara Ugolini","doi":"10.1016/j.apal.2023.103391","DOIUrl":"10.1016/j.apal.2023.103391","url":null,"abstract":"<div><p>In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of <span><math><mi>MTL</mi></math></span>-algebras, i.e., bounded commutative integral residuated lattices generated by chains.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007223001483/pdfft?md5=566d39a3ec46ed86a39bb7490723fd1b&pid=1-s2.0-S0168007223001483-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-11DOI: 10.1016/j.apal.2023.103402
JinHoo Ahn , Joonhee Kim
In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP2 can be witnessed by a formula with a tree of tuples holding ‘arbitrary homogeneous inconsistency’ (e.g., weak k-TP1 conditions or other possible inconsistency configurations).
And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP1, and by using this, we investigate the problem of (in)equality of SOP1 and SOP2.
Assuming the existence of a formula having SOP1 such that no finite conjunction of it has SOP2, we observe that the formula must witness some tree-property-like phenomenon, which we will call the antichain tree property (ATP, see Definition 4.1). We show that ATP implies SOP1 and TP2, but the converse of each implication does not hold. So the class of NATP theories (theories without ATP) contains the class of NSOP1 theories and the class of NTP2 theories.
At the end of the paper, we construct a structure whose theory has a formula having ATP, but any conjunction of the formula does not have SOP2. So this example shows that SOP1 and SOP2 are not the same at the level of formulas, i.e., there is a formula having SOP1, while any finite conjunction of it does not witness SOP2 (but a variation of the formula still has SOP2).
{"title":"SOP1, SOP2, and antichain tree property","authors":"JinHoo Ahn , Joonhee Kim","doi":"10.1016/j.apal.2023.103402","DOIUrl":"10.1016/j.apal.2023.103402","url":null,"abstract":"<div><p>In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP<sub>2</sub> can be witnessed by a formula with a tree of tuples holding ‘arbitrary homogeneous inconsistency’ (e.g., weak <em>k</em>-TP<sub>1</sub> conditions or other possible inconsistency configurations).</p><p>And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP<sub>1</sub>, and by using this, we investigate the problem of (in)equality of SOP<sub>1</sub> and SOP<sub>2</sub>.</p><p>Assuming the existence of a formula having SOP<sub>1</sub> such that no finite conjunction of it has SOP<sub>2</sub><span>, we observe that the formula must witness some tree-property-like phenomenon, which we will call the antichain tree property (ATP, see </span><span>Definition 4.1</span>). We show that ATP implies SOP<sub>1</sub> and TP<sub>2</sub>, but the converse of each implication does not hold. So the class of NATP theories (theories without ATP) contains the class of NSOP<sub>1</sub> theories and the class of NTP<sub>2</sub> theories.</p><p>At the end of the paper, we construct a structure whose theory has a formula having ATP, but any conjunction of the formula does not have SOP<sub>2</sub>. So this example shows that SOP<sub>1</sub> and SOP<sub>2</sub> are not the same at the level of formulas, i.e., there is a formula having SOP<sub>1</sub>, while any finite conjunction of it does not witness SOP<sub>2</sub> (but a variation of the formula still has SOP<sub>2</sub>).</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138575042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-07DOI: 10.1016/j.apal.2023.103392
Richard Matthews , Michael Rathjen
We study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over Intuitionistic Kripke-Platek set theory without Infinity. Following this, we investigate when L can fail to be an inner model in the traditional sense. Namely, we show that over Constructive Zermelo-Fraenkel (even with the Power Set axiom) one cannot prove that the Axiom of Exponentiation holds in L.
我们研究直觉主义理论的可构造宇宙 L 的性质。我们给出了一个基本运算的扩展集,它足以在直观克里普克-普拉特克集合论上生成无穷大的宇宙。在此基础上,我们研究了什么情况下 L 不能成为传统意义上的内部模型。也就是说,我们证明在构造泽梅洛-弗兰克尔(即使有幂集公理)上,我们无法证明幂级数公理在 L 中成立。
{"title":"Constructing the constructible universe constructively","authors":"Richard Matthews , Michael Rathjen","doi":"10.1016/j.apal.2023.103392","DOIUrl":"10.1016/j.apal.2023.103392","url":null,"abstract":"<div><p>We study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over Intuitionistic Kripke-Platek set theory without Infinity. Following this, we investigate when L can fail to be an inner model in the traditional sense. Namely, we show that over Constructive Zermelo-Fraenkel (even with the Power Set axiom) one cannot prove that the Axiom of Exponentiation holds in L.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138548019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-20DOI: 10.1016/j.apal.2023.103390
Samuele Maschio, Davide Trotta
Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras, accompanied by various concrete examples. Then, we shift our focus to viewing implicative algebras as a generalization of partial combinatory algebras. We abstract the notion of a category of assemblies, partition assemblies, and modest sets to arbitrary implicative algebras, and thoroughly investigate their categorical properties and interrelationships.
{"title":"On categorical structures arising from implicative algebras: From topology to assemblies","authors":"Samuele Maschio, Davide Trotta","doi":"10.1016/j.apal.2023.103390","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103390","url":null,"abstract":"<div><p>Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras, accompanied by various concrete examples. Then, we shift our focus to viewing implicative algebras as a generalization of partial combinatory algebras. We abstract the notion of a category of assemblies, partition assemblies, and modest sets to arbitrary implicative algebras, and thoroughly investigate their categorical properties and interrelationships.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007223001471/pdfft?md5=e62dfd1384a82cf70c38bd805eb49847&pid=1-s2.0-S0168007223001471-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138484884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-14DOI: 10.1016/j.apal.2023.103389
Dragan Doder , Zoran Ognjanović
This work presents a proof-theoretical and model-theoretical approach to probabilistic temporal logic. We present two novel logics; each of them extends both the language of linear time logic (LTL) and the language of probabilistic logic with polynomial weight formulas. The first logic is designed for reasoning about probabilities of temporal events, allowing statements like “the probability that A will hold in next moment is at least the probability that B will always hold” and conditional probability statements like “probability that A will always hold, given that B holds, is at least one half”, where A and B are arbitrary statements. We axiomatize this logic, provide corresponding sigma additive semantics and prove that the axiomatization is sound and strongly complete. We show that the satisfiability problem for our logic is decidable, by presenting a procedure which runs in polynomial space. We also present a logic with much richer language, in which probabilities are not attached only to temporal events, but the language allows arbitrary nesting of probability and temporal operators, allowing statements like “probability that tomorrow the chance of rain will be less than 80% is at least a half”. For this logic we prove a decidability result.
这项研究提出了一种证明理论和模型理论方法来研究概率时态逻辑。我们提出了两个新颖的逻辑;每个逻辑都扩展了线性时间逻辑(LTL)语言和具有多项式权重公式的概率逻辑语言。第一种逻辑是为推理时间事件的概率而设计的,它允许 "A 在下一时刻成立的概率至少是 B 始终成立的概率 "这样的语句,以及 "A 始终成立的概率至少是 B 成立的二分之一 "这样的条件概率语句,其中 A 和 B 是任意语句。我们对这一逻辑进行了公理化,提供了相应的西格玛加法语义,并证明了公理化的合理性和强完备性。我们提出了一个在多项式空间内运行的过程,从而证明我们逻辑的可满足性问题是可解的。我们还提出了一种语言更为丰富的逻辑,在这种逻辑中,概率不仅与时间事件相关联,而且允许概率和时间运算符的任意嵌套,允许 "明天下雨的概率小于 80% 的概率至少是一半 "这样的语句。对于这种逻辑,我们证明了一个可解性结果。
{"title":"Probabilistic temporal logic with countably additive semantics","authors":"Dragan Doder , Zoran Ognjanović","doi":"10.1016/j.apal.2023.103389","DOIUrl":"10.1016/j.apal.2023.103389","url":null,"abstract":"<div><p>This work presents a proof-theoretical and model-theoretical approach to probabilistic temporal logic. We present two novel logics; each of them extends both the language of linear time logic (LTL) and the language of probabilistic logic with polynomial weight formulas. The first logic is designed for reasoning about probabilities of temporal events, allowing statements like “the probability that A will hold in next moment is at least the probability that B will always hold” and conditional probability statements like “probability that A will always hold, given that B holds, is at least one half”, where A and B are arbitrary statements. We axiomatize this logic, provide corresponding sigma additive semantics and prove that the axiomatization is sound and strongly complete. We show that the satisfiability problem for our logic is decidable, by presenting a procedure which runs in polynomial space. We also present a logic with much richer language, in which probabilities are not attached only to temporal events, but the language allows arbitrary nesting of probability and temporal operators, allowing statements like “probability that tomorrow the chance of rain will be less than 80% is at least a half”. For this logic we prove a decidability result.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S016800722300146X/pdfft?md5=b48bdc121fa115161627545db4c861fd&pid=1-s2.0-S016800722300146X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135716247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.1016/j.apal.2023.103388
Sam van Gool, Jérémie Marquès
This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness properties, and allows for a modular, syntax-free treatment of these model-theoretic results. As further applications of the same method, we prove completeness theorems for constant domain and Gödel-Dummett intuitionistic predicate logics.
{"title":"On duality and model theory for polyadic spaces","authors":"Sam van Gool, Jérémie Marquès","doi":"10.1016/j.apal.2023.103388","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103388","url":null,"abstract":"<div><p>This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness properties, and allows for a modular, syntax-free treatment of these model-theoretic results. As further applications of the same method, we prove completeness theorems for constant domain and Gödel-Dummett intuitionistic predicate logics.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134656886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.1016/j.apal.2023.103387
Athar Abdul-Quader , Mateusz Łełyk
We study subsets of countable recursively saturated models of which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets X such that there is a satisfaction class S where S behaves correctly on an idempotent disjunction of length c if and only if . We generalize this result to characterize several types of pathologies including double negations, blocks of extraneous quantifiers, and binary disjunctions and conjunctions. We find a surprising relationship between the cuts which can be defined in this way and arithmetic saturation: namely, a countable nonstandard model is arithmetically saturated if and only if every cut can be the “idempotent disjunctively correct cut” in some satisfaction class. We describe the relationship between types of pathologies and the closure properties of the cuts defined by these pathologies.
{"title":"Pathologies in satisfaction classes","authors":"Athar Abdul-Quader , Mateusz Łełyk","doi":"10.1016/j.apal.2023.103387","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103387","url":null,"abstract":"<div><p>We study subsets of countable recursively saturated models of <span><math><mi>PA</mi></math></span> which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets <em>X</em> such that there is a satisfaction class <em>S</em> where <em>S</em> behaves correctly on an idempotent disjunction of length <em>c</em> if and only if <span><math><mi>c</mi><mo>∈</mo><mi>X</mi></math></span>. We generalize this result to characterize several types of pathologies including double negations, blocks of extraneous quantifiers, and binary disjunctions and conjunctions. We find a surprising relationship between the cuts which can be defined in this way and arithmetic saturation: namely, a countable nonstandard model is arithmetically saturated if and only if every cut can be the “idempotent disjunctively correct cut” in some satisfaction class. We describe the relationship between types of pathologies and the closure properties of the cuts defined by these pathologies.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49725035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-06DOI: 10.1016/j.apal.2023.103386
M. Shahryari
It is known that an algebra is geometrically equivalent to any of its filterpowers if it is -compact. We present an explicit description for the radicals of systems of equation over an algebra A and then we prove the above assertion by an elementary new argument. Then we define -compact algebras and κ-filterpowers for any infinite cardinal κ. We show that any -compact algebra is geometric equivalent to its κ-filterpowers. As there is no algebraic description of the κ-quasivariety generated by an algebra, the classical argument can not be applied in this case, while our proof still works.
{"title":"On the geometric equivalence of algebras","authors":"M. Shahryari","doi":"10.1016/j.apal.2023.103386","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103386","url":null,"abstract":"<div><p>It is known that an algebra is geometrically equivalent to any of its filterpowers if it is <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-compact. We present an explicit description for the radicals of systems of equation over an algebra <em>A</em> and then we prove the above assertion by an elementary new argument. Then we define <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span>-compact algebras and <em>κ</em>-filterpowers for any infinite cardinal <em>κ</em>. We show that any <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span>-compact algebra is geometric equivalent to its <em>κ</em>-filterpowers. As there is no algebraic description of the <em>κ</em>-quasivariety generated by an algebra, the classical argument can not be applied in this case, while our proof still works.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49725032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1016/j.apal.2023.103302
Ram Sewak Dubey , Giorgio Laguzzi
Social welfare relations satisfying Pareto and equity principles on infinite utility streams have revealed a non-constructive nature, specifically by showing that in general they imply the existence of non-Ramsey sets and non-Lebesgue measurable sets. In [4, Problem 11.14], the authors ask whether such a connection holds with non-Baire sets as well. In this paper we answer such a question showing that several versions of Pareto principles acting on different utility domains imply the existence of non-Baire sets. Furthermore, we analyze in more details the needed fragments of AC and we start a systematic investigation of a social welfare diagram in a similar fashion done in the past decades concerning cardinal invariants and regularity properties of the reals. In doing that we use tools from forcing theory, such as specific tree-forcings (in particular variants of Silver and Mathias forcings) and Shelah's amalgamation.
{"title":"Social welfare relations and irregular sets","authors":"Ram Sewak Dubey , Giorgio Laguzzi","doi":"10.1016/j.apal.2023.103302","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103302","url":null,"abstract":"<div><p>Social welfare relations satisfying Pareto and equity principles on infinite utility streams have revealed a non-constructive nature, specifically by showing that in general they imply the existence of non-Ramsey sets and non-Lebesgue measurable sets. In <span>[4, Problem 11.14]</span>, the authors ask whether such a connection holds with non-Baire sets as well. In this paper we answer such a question showing that several versions of Pareto principles acting on different utility domains imply the existence of non-Baire sets. Furthermore, we analyze in more details the needed fragments of AC and we start a systematic investigation of a <em>social welfare diagram</em> in a similar fashion done in the past decades concerning cardinal invariants and regularity properties of the reals. In doing that we use tools from forcing theory, such as specific tree-forcings (in particular variants of Silver and Mathias forcings) and Shelah's amalgamation.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}