ABSTRACT This paper is concerned with the existence of nontrivial weak solutions for a class of p(x)-Laplacian problem under Steklov boundary condition. The variable exponent theory of generalized Lebesgue-Sobolev spaces and the concentration-compactness principle for weighted variable exponent spaces are used for this purpose.
{"title":"A Critical p(x)-Laplacian Steklov Type Problem with Weights","authors":"Mostafa Allaoui, O. Darhouche","doi":"10.1515/ms-2023-0109","DOIUrl":"https://doi.org/10.1515/ms-2023-0109","url":null,"abstract":"ABSTRACT This paper is concerned with the existence of nontrivial weak solutions for a class of p(x)-Laplacian problem under Steklov boundary condition. The variable exponent theory of generalized Lebesgue-Sobolev spaces and the concentration-compactness principle for weighted variable exponent spaces are used for this purpose.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139021425","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}
ABSTRACT Let X, Y be two Banach spaces over K=ℝ [mathbb{K}=mathbb{R}] or ℂ [mathbb{C}] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.
ABSTRACT Let X, Y be two Banach spaces over K=ℝ [mathbb{K}=mathbb{R}] or ℂ [mathbb{C}] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C1-compact operator.我们提供了充分条件来断言扰动算子 f 是 C1-差分。当其中一个条件不成立，而 y 是一个正则值时，方程 f(x) = y 最多有有限个解。作为主要结果的结果，给出了两个推论。第二个定理研究的是有限维情况。作为应用，给出了一个例子。我们结果的证明基于弗雷德霍姆算子的性质以及局部和全局逆映射定理。
{"title":"Operators that are Weakly Coercive and a Compact Perturbation","authors":"J. M. Soriano Arbizu, Manuel Ordoñez Cabrera","doi":"10.1515/ms-2023-0112","DOIUrl":"https://doi.org/10.1515/ms-2023-0112","url":null,"abstract":"ABSTRACT Let X, Y be two Banach spaces over K=ℝ [mathbb{K}=mathbb{R}] or ℂ [mathbb{C}] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138991853","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}
Ariane G. Tallee Kakeu, L. Strüngmann, B. B. Koguep Njionou, Celestin Lele
ABSTRACT In this paper, we provide a new characterization of ℒ-fuzzy ideals of residuated lattices, which allows us to describe ℒ-fuzzy ideals generated by ℒ-fuzzy sets. Thanks to the latter, we endow the lattice of ℒ-fuzzy ideals of a residuated lattice with suitable operations. Moreover, we introduce the notion of ℒ-fuzzy annihilator of an ℒ-fuzzy subset of a residuated lattice with respect to an ℒ-fuzzy ideal and investigate some of its properties. To this extent, we show that the set of all ℒ-fuzzy ideals of a residuated lattice is a complete Heyting algebra. Furthermore, we define some types of ℒ-fuzzy ideals of residuated lattices, namely stable ℒ-fuzzy ideals relative to an ℒ-fuzzy set, and involutory ℒ-fuzzy ideals relative to an ℒ-fuzzy ideal. Finally, we prove that the set of all stable ℒ-fuzzy ideals relative to an ℒ-fuzzy set is also a complete Heyting algebra, and that the set of involutory ℒ-fuzzy ideals relative to an ℒ-fuzzy ideal is a complete Boolean algebra.
{"title":"ℒ-fuzzy Annihilators in Residuated Lattices","authors":"Ariane G. Tallee Kakeu, L. Strüngmann, B. B. Koguep Njionou, Celestin Lele","doi":"10.1515/ms-2023-0098","DOIUrl":"https://doi.org/10.1515/ms-2023-0098","url":null,"abstract":"ABSTRACT In this paper, we provide a new characterization of ℒ-fuzzy ideals of residuated lattices, which allows us to describe ℒ-fuzzy ideals generated by ℒ-fuzzy sets. Thanks to the latter, we endow the lattice of ℒ-fuzzy ideals of a residuated lattice with suitable operations. Moreover, we introduce the notion of ℒ-fuzzy annihilator of an ℒ-fuzzy subset of a residuated lattice with respect to an ℒ-fuzzy ideal and investigate some of its properties. To this extent, we show that the set of all ℒ-fuzzy ideals of a residuated lattice is a complete Heyting algebra. Furthermore, we define some types of ℒ-fuzzy ideals of residuated lattices, namely stable ℒ-fuzzy ideals relative to an ℒ-fuzzy set, and involutory ℒ-fuzzy ideals relative to an ℒ-fuzzy ideal. Finally, we prove that the set of all stable ℒ-fuzzy ideals relative to an ℒ-fuzzy set is also a complete Heyting algebra, and that the set of involutory ℒ-fuzzy ideals relative to an ℒ-fuzzy ideal is a complete Boolean algebra.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139018457","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}
ABSTRACT Let k ≥ 2. A generalization of the well-known Pell sequence is the k-Pell sequence whose first k terms are 0,…, 0, 1 and each term afterwards is given by the linear recurrence pn(k)=2Pn−1(k)+Pn−2(k)+⋯+Pn−k(k). The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k-Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [Repdigits in generalized Pell sequences, Arch. Math. (Brno) 56(4) (2020), 249–262].
摘要 设 k ≥ 2。众所周知的佩尔序列的广义化是 k-Pell 序列，其前 k 项为 0，...，0，1，之后的每项由线性递推公式 pn(k)=2Pn-1(k)+Pn-2(k)+⋯+Pn-k(k)给出。本文的目的是证明 11、33、55、88 和 99 都是可以用两个 k-Pell 的和或差来表示的重数字。我们主要定理的证明使用了代数数对数线性形式的下界，以及改进版的贝克-达文波特还原法（由杜杰拉和佩索提出）。这扩展了布拉沃和埃雷拉 [Repdigits in generalized Pell sequences, Arch.(Brno) 56(4) (2020), 249-262].
{"title":"On Repdigits Which are Sums or Differences of Two k-Pell Numbers","authors":"Mariama Ndao Faye, S. Rihane, A. Togbé","doi":"10.1515/ms-2023-0102","DOIUrl":"https://doi.org/10.1515/ms-2023-0102","url":null,"abstract":"ABSTRACT Let k ≥ 2. A generalization of the well-known Pell sequence is the k-Pell sequence whose first k terms are 0,…, 0, 1 and each term afterwards is given by the linear recurrence pn(k)=2Pn−1(k)+Pn−2(k)+⋯+Pn−k(k). The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k-Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [Repdigits in generalized Pell sequences, Arch. Math. (Brno) 56(4) (2020), 249–262].","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139013086","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}
ABSTRACT In this paper, we consider a first-order iterative differential inclusion. We study the existence of solutions and some topological proprieties of the attainable set, where the right hand side is an upper semi-continuous multifunction with convex values. Then, we treat the autonomous problem under assumptions that do not require the convexity of the values and that weaken the assumption on the upper semi-continuity.
{"title":"On the Attainable Set of Iterative Differential Inclusions","authors":"Samia Ghalia, Doria Affane","doi":"10.1515/ms-2023-0107","DOIUrl":"https://doi.org/10.1515/ms-2023-0107","url":null,"abstract":"ABSTRACT In this paper, we consider a first-order iterative differential inclusion. We study the existence of solutions and some topological proprieties of the attainable set, where the right hand side is an upper semi-continuous multifunction with convex values. Then, we treat the autonomous problem under assumptions that do not require the convexity of the values and that weaken the assumption on the upper semi-continuity.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139018610","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}
ABSTRACT We address Peirce decompositions for evolution algebras at idempotent elements contained in the associative nucleus of the evolution algebras. If the idempotent elements are natural vectors, the requirement of being nuclear is then proved to be equivalent to the evolution algebra to be baric. A description of baric evolution algebras is also provided.
{"title":"Peirce Decompositions for Evolution Algebras","authors":"I. Paniello","doi":"10.1515/ms-2023-0103","DOIUrl":"https://doi.org/10.1515/ms-2023-0103","url":null,"abstract":"ABSTRACT We address Peirce decompositions for evolution algebras at idempotent elements contained in the associative nucleus of the evolution algebras. If the idempotent elements are natural vectors, the requirement of being nuclear is then proved to be equivalent to the evolution algebra to be baric. A description of baric evolution algebras is also provided.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139020747","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}
ABSTRACT We generalize cyclic Jensen’s inequality utilizing the theory of 4-convex function under the effect of introduced Green functions. We formulate result for power means and quasi means. Also we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi-divergence, Shannon-entropy, Kullback-Leibler divergence and χ2-divergence.
{"title":"New Bounds of Cyclic Jensen’s Differences via Weighted Hadamard Inequalities with Applications","authors":"T. Rasheed, S. Butt, Ɖilda Pečarić, J. Pečarić","doi":"10.1515/ms-2023-0105","DOIUrl":"https://doi.org/10.1515/ms-2023-0105","url":null,"abstract":"ABSTRACT We generalize cyclic Jensen’s inequality utilizing the theory of 4-convex function under the effect of introduced Green functions. We formulate result for power means and quasi means. Also we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi-divergence, Shannon-entropy, Kullback-Leibler divergence and χ2-divergence.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139026128","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}
ABSTRACT This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.
{"title":"Some Approximation Properties of Operators Including Degenerate Appell Polynomials","authors":"Bilge Zehra Sergi, Gürhan Içöz, Bayram Çekim","doi":"10.1515/ms-2023-0111","DOIUrl":"https://doi.org/10.1515/ms-2023-0111","url":null,"abstract":"ABSTRACT This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139015765","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}
J. M. Cornejo, Michael Kinyon, H. P. Sankappanavar
ABSTRACT In 1973, Katriňák proved that regular double p-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák’s theorem, in the sense that in the variety ℝDℙℂℍ [mathbb{R}mathbb{D}mathbb{P}mathbb{C}mathbb{H}] of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák’s formula. As applications of this result together with the above-mentioned Katriňák’s theorem, we show that the varieties ℝDBLℙ [mathbb{R}mathbb{D}mathbb{B}mathbb{L}mathbb{P}] , ℝDℙℂℍ [mathbb{R}mathbb{D}mathbb{P}mathbb{C}mathbb{H}] , ℝℙℂℍd [mathbb{R}mathbb{P}mathbb{C}{{mathbb{H}}^{d}}] and ℝDBLℍ [mathbb{R}mathbb{D}mathbb{B}mathbb{L}mathbb{H}] of regular double p-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties ℝDMℙ [mathbb{R}mathbb{D}mathbb{M}mathbb{P}] , ℝDMℍ [mathbb{R}mathbb{D}mathbb{M}mathbb{H}] , ℝDMDBLℍ [mathbb{R}mathbb{D}mathbb{M}mathbb{D}mathbb{B}mathbb{L}mathbb{H}] , ℝDMDBLℙ [mathbb{R}mathbb{D}mathbb{M}mathbb{D}mathbb{B}mathbb{L}mathbb{P}] of regular De Morgan p-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double p-algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double p-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2ℵ0 [{{2}^{{{aleph }_{0}}}}] . We then define new logics, ℛDPCℋ [mathcal{R}mathcal{D}mathcal{P}mathcal{C}mathcal{H}] , ℛPCℋd [mathcal{R}mathcal{P}mathcal{C}{{mathcal{H}}^{d}}] and ℛDℳℋ [mathcal{R}mathcal{D}mathcal{M}mathcal{H}] , and show that they are algebraizable with ℝDℙℂℍ [mathbb{R}mathbb{D}mathbb{P}mathbb{C}mathbb{H}] , ℝℙℂℍd [mathbb{R}mathbb{P}mathbb{C}{{mathbb{H}}^{d}}] and ℝDMℍ [mathbb{R}mathbb{D}mathbb{M}mathbb{H}] , respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality 2ℵ0 [{{2}^{{{aleph }_{0}}}}] .
摘要 1973 年，卡特里纳克通过巧妙地用伪补数及其对偶来构造海廷蕴涵及其对偶的二元项，证明了正则双 p 格拉斯可视为（正则）双海廷格拉斯。在本文中，我们证明了卡特里纳克定理的逆定理，即在正则双伪补齐海廷代数的ℝDℙℂℍ [mathbb{R}mathbb{D}mathbb{P}mathbb{C}mathbb{H}]中，蕴涵运算 → 满足卡特里纳克公式。作为这一结果与上述卡特里纳克定理的应用、我们证明了ℝDBLℙ [mathbb{R}mathbb{D}mathbb{B}mathbb{L}mathbb{P}] , ℝDℙℂℍ [mathbb{R}mathbb{D}mathbb{P}mathbb{C}mathbb{H}] 、ℝℙℂℍd [mathbb{R}mathbb{P}mathbb{C}{{mathbb{H}}^{d}] and ℝDBLℍ [mathbb{R}mathbb{D}mathbb{B}mathbb{L}mathbb{H}}] of regular double p-algebras、正则双伪互补海廷布拉斯、正则伪互补双海廷布拉斯和正则双海廷布拉斯、分别是项等价的，而且ℝDMℙ [mathbb{R}mathbb{D}mathbb{M}mathbb{P}] 、ℝDMℍ [mathbb{R}mathbb{D}mathbb{M}mathbb{H}] , ℝDMDBLℍ [mathbb{R}mathbb{D}mathbb{M}mathbb{D}mathbb{B}mathbb{L}mathbb{H}] 、ℝDMDBLℙ [mathbb{R}mathbb{D}mathbb{M}mathbb{D}mathbb{B}mathbb{L}mathbb{P}] of regular De Morgan p-algebras、regular De Morgan Heyting algebras、regular De Morgan double Heyting algebras 和 regular De Morgan double p-algebras，也是项等价的。根据这些结果，以及亚当斯、桑卡帕纳瓦尔和瓦兹-德-卡瓦略最近关于正则双 p-格拉斯和正则伪补德-摩根格拉斯的研究成果，我们推导出所有这些变体的子变体的晶格都有 cardinality 2ℵ0 [{{2}^{{{aleph }_{0}}}}] 。然后我们定义新的逻辑，ℛDPCℋ [mathcal{R}mathcal{D}mathcal{P}mathcal{C}mathcal{H}] 、ℛPCℋd [mathcal{R}mathcal{P}mathcal{C}{mathcal{H}}^{d}] and ℛDℳℋ [mathcal{R}mathcal{D}mathcal{M}mathcal{H}] 、并证明它们与ℝDℙℂℍ [mathbb{R}mathbb{D}mathbb{P}mathbb{C}mathbb{H}] 是可代数的、和ℝDMℂℍd [（mathbb{R}mathbb{D}mathbb{M}mathbb{H}^{d}}]，分别作为它们等价的代数语义。我们还可以推导出，上述所有逻辑的扩展晶格都有 cardinality 2ℵ0 [{{2}^{{aleph }_{0}}}}] 。
{"title":"Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications","authors":"J. M. Cornejo, Michael Kinyon, H. P. Sankappanavar","doi":"10.1515/ms-2023-0099","DOIUrl":"https://doi.org/10.1515/ms-2023-0099","url":null,"abstract":"ABSTRACT In 1973, Katriňák proved that regular double p-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák’s theorem, in the sense that in the variety ℝDℙℂℍ [mathbb{R}mathbb{D}mathbb{P}mathbb{C}mathbb{H}] of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák’s formula. As applications of this result together with the above-mentioned Katriňák’s theorem, we show that the varieties ℝDBLℙ [mathbb{R}mathbb{D}mathbb{B}mathbb{L}mathbb{P}] , ℝDℙℂℍ [mathbb{R}mathbb{D}mathbb{P}mathbb{C}mathbb{H}] , ℝℙℂℍd [mathbb{R}mathbb{P}mathbb{C}{{mathbb{H}}^{d}}] and ℝDBLℍ [mathbb{R}mathbb{D}mathbb{B}mathbb{L}mathbb{H}] of regular double p-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties ℝDMℙ [mathbb{R}mathbb{D}mathbb{M}mathbb{P}] , ℝDMℍ [mathbb{R}mathbb{D}mathbb{M}mathbb{H}] , ℝDMDBLℍ [mathbb{R}mathbb{D}mathbb{M}mathbb{D}mathbb{B}mathbb{L}mathbb{H}] , ℝDMDBLℙ [mathbb{R}mathbb{D}mathbb{M}mathbb{D}mathbb{B}mathbb{L}mathbb{P}] of regular De Morgan p-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double p-algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double p-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2ℵ0 [{{2}^{{{aleph }_{0}}}}] . We then define new logics, ℛDPCℋ [mathcal{R}mathcal{D}mathcal{P}mathcal{C}mathcal{H}] , ℛPCℋd [mathcal{R}mathcal{P}mathcal{C}{{mathcal{H}}^{d}}] and ℛDℳℋ [mathcal{R}mathcal{D}mathcal{M}mathcal{H}] , and show that they are algebraizable with ℝDℙℂℍ [mathbb{R}mathbb{D}mathbb{P}mathbb{C}mathbb{H}] , ℝℙℂℍd [mathbb{R}mathbb{P}mathbb{C}{{mathbb{H}}^{d}}] and ℝDMℍ [mathbb{R}mathbb{D}mathbb{M}mathbb{H}] , respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality 2ℵ0 [{{2}^{{{aleph }_{0}}}}] .","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138993768","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}
ABSTRACT We extend the notion of local connectedness and uniform local connectedness to the category MσFrm of metric σ-frames. We provide the construction of the uniformly locally connected reflection of a locally connected metric σ-frame. We also discuss complete metric σ-frames and show the existence of a completion for a metric σ-frame in the category MσFrm.
{"title":"Uniform Local Connectedness and Completion of Metric σ-Frames","authors":"I. Naidoo","doi":"10.1515/ms-2023-0100","DOIUrl":"https://doi.org/10.1515/ms-2023-0100","url":null,"abstract":"ABSTRACT We extend the notion of local connectedness and uniform local connectedness to the category MσFrm of metric σ-frames. We provide the construction of the uniformly locally connected reflection of a locally connected metric σ-frame. We also discuss complete metric σ-frames and show the existence of a completion for a metric σ-frame in the category MσFrm.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139025165","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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