Pub Date : 2022-10-14DOI: 10.18778/0138-0680.2022.09
Batoul Ganji Saffar, R. Borzooei, M. Kologani
In this paper, we apply the notion of (n)-fold filters to the (EQ)-algebras and introduce the concepts of (n)-fold positive implicative (implicative, obstinate, fantastic) (pre)filter on an (EQ)-algebra (mathcal{E}). Then we investigate some properties and relations among them. We prove that the quotient structure (mathcal{E}/F) that is made by an 1-fold positive implicative filter of an (EQ)-algebra (mathcal{E}) is a good (EQ)-algebra and the quotient structure (mathcal{E}/F) that is made by an 1-fold fantastic filter of a good (EQ)-algebra (mathcal{E}) is an (IEQ)-algebra.
{"title":"(n)-fold filters of EQ-algebras","authors":"Batoul Ganji Saffar, R. Borzooei, M. Kologani","doi":"10.18778/0138-0680.2022.09","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.09","url":null,"abstract":"In this paper, we apply the notion of (n)-fold filters to the (EQ)-algebras and introduce the concepts of (n)-fold positive implicative (implicative, obstinate, fantastic) (pre)filter on an (EQ)-algebra (mathcal{E}). Then we investigate some properties and relations among them. We prove that the quotient structure (mathcal{E}/F) that is made by an 1-fold positive implicative filter of an (EQ)-algebra (mathcal{E}) is a good (EQ)-algebra and the quotient structure (mathcal{E}/F) that is made by an 1-fold fantastic filter of a good (EQ)-algebra (mathcal{E}) is an (IEQ)-algebra.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46725470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-20DOI: 10.18778/0138-0680.2022.18
Majid Alizadeh, Masoud Memarzadeh
In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property.
{"title":"Interpolation Property on Visser's Formal Propositional Logic","authors":"Majid Alizadeh, Masoud Memarzadeh","doi":"10.18778/0138-0680.2022.18","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.18","url":null,"abstract":" In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48909355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-14DOI: 10.18778/0138-0680.2022.13
P. Yiarayong
The aim of this manuscript is to introduce the ((alpha,beta))-hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of ((alpha,beta))-hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study ((alpha,beta))-hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of ((alpha,beta))-hesitant fuzzy ideal generated by a hesitant fuzzy ideal in a semigroup. In addition, we also introduce the concepts of ((alpha,beta))-hesitant fuzzy semiprime sets of semigroups, and characterize regular semigroups in terms of ((alpha,beta))-hesitant fuzzy left ideals and ((alpha,beta))-hesitant fuzzy right ideals. Finally, several characterizations of regular and intra-regular semigroups by the properties of ((alpha,beta))-hesitant ideals are given.
{"title":"An (α,β)-Hesitant Fuzzy Set Approach to Ideal Theory in Semigroups","authors":"P. Yiarayong","doi":"10.18778/0138-0680.2022.13","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.13","url":null,"abstract":"The aim of this manuscript is to introduce the ((alpha,beta))-hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of ((alpha,beta))-hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study ((alpha,beta))-hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of ((alpha,beta))-hesitant fuzzy ideal generated by a hesitant fuzzy ideal in a semigroup. In addition, we also introduce the concepts of ((alpha,beta))-hesitant fuzzy semiprime sets of semigroups, and characterize regular semigroups in terms of ((alpha,beta))-hesitant fuzzy left ideals and ((alpha,beta))-hesitant fuzzy right ideals. Finally, several characterizations of regular and intra-regular semigroups by the properties of ((alpha,beta))-hesitant ideals are given.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44823085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-11DOI: 10.18778/0138-0680.2023.23
M. Coniglio, Guilherme V. Toledo
It is well known that there is a correspondence between sets and complete, atomic Boolean algebras ((textit{CABA})s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of (textbf{Set}) and the category of (textit{CABA})s. We modify this result by taking multialgebras over a signature (Sigma), specifically those whose non-deterministic operations cannot return the empty-set, to (textit{CABA})s with their zero element removed (which we call a ({em bottomless Boolean algebra})) equipped with a structure of (Sigma)-algebra compatible with its order (that we call ({em ord-algebras})). Conversely, an ord-algebra over (Sigma) is taken to its set of atomic elements equipped with a structure of multialgebra over (Sigma). This leads to an equivalence between the category of (Sigma)-multialgebras and the category of ord-algebras over (Sigma). The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.
众所周知，集合和完整的原子布尔代数（（textit｛CABA｝）s）之间存在对应关系，将集合作为其幂集，反之，将完整的原子Boolean代数作为其原子元素集。当然，这种对应关系导致了（textbf｛Set｝）的相反类别和（text it｛CABA｝，到（textit｛CABA｝）s，它们的零元素被移除（我们称之为（｛em无底布尔代数）），配备有与其阶兼容的（ Sigma）-代数结构（我们称为（ em ord代数））。相反，在（西格玛）上的一个ord代数被带到它的原子元素集，该原子元素集配备了在（西格玛）之上的多代数的结构。这导致了（ Sigma）-多代数的范畴和（西格玛）上的ord代数的范畴之间的等价性。这里的直觉是，如果人们希望这样做，非决定论可能会被底层结构的足够丰富的排序所取代。
{"title":"A Category of Ordered Algebras Equivalent to the Category of Multialgebras","authors":"M. Coniglio, Guilherme V. Toledo","doi":"10.18778/0138-0680.2023.23","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.23","url":null,"abstract":"It is well known that there is a correspondence between sets and complete, atomic Boolean algebras ((textit{CABA})s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of (textbf{Set}) and the category of (textit{CABA})s.\u0000We modify this result by taking multialgebras over a signature (Sigma), specifically those whose non-deterministic operations cannot return the empty-set, to (textit{CABA})s with their zero element removed (which we call a ({em bottomless Boolean algebra})) equipped with a structure of (Sigma)-algebra compatible with its order (that we call ({em ord-algebras})). Conversely, an ord-algebra over (Sigma) is taken to its set of atomic elements equipped with a structure of multialgebra over (Sigma). This leads to an equivalence between the category of (Sigma)-multialgebras and the category of ord-algebras over (Sigma).\u0000The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42373194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-09DOI: 10.18778/0138-0680.2022.10
R. Borzooei, Elham Babaei
When it comes to making decisions in vague problems, rough is one of the best tools to help analyzers. So based on rough and hoop concepts, two kinds of approximations (Lower and Upper) for filters in hoops are defined, and then some properties of them are investigated by us. We prove that these approximations- lower and upper- are interior and closure operators, respectively. Also after defining a hyper operation in hoops, we show that by using this hyper operation, set of all rough filters is monoid. For more study, we define the implicative operation on the set of all rough filters and prove that this set with implication and intersection is made a hoop.
{"title":"Constructing a Hoop Using Rough Filters","authors":"R. Borzooei, Elham Babaei","doi":"10.18778/0138-0680.2022.10","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.10","url":null,"abstract":"When it comes to making decisions in vague problems, rough is one of the best tools to help analyzers. So based on rough and hoop concepts, two kinds of approximations (Lower and Upper) for filters in hoops are defined, and then some properties of them are investigated by us. We prove that these approximations- lower and upper- are interior and closure operators, respectively. Also after defining a hyper operation in hoops, we show that by using this hyper operation, set of all rough filters is monoid. For more study, we define the implicative operation on the set of all rough filters and prove that this set with implication and intersection is made a hoop.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43697119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-09DOI: 10.18778/0138-0680.2022.16
S. I. Bashmakov, Tatyana Yu. Zvereva
This paper proposes a semantic description of the linear step-like temporal multi-agent logic with the universal modality (mathcal{LTK}.sl_U) based on the idea of non-reflexive non-transitive nature of time. We proved a finite model property and projective unification for this logic.
{"title":"Unification and Finite Model Property for Linear Step-Like Temporal Multi-Agent Logic with the Universal Modality","authors":"S. I. Bashmakov, Tatyana Yu. Zvereva","doi":"10.18778/0138-0680.2022.16","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.16","url":null,"abstract":"This paper proposes a semantic description of the linear step-like temporal multi-agent logic with the universal modality (mathcal{LTK}.sl_U) based on the idea of non-reflexive non-transitive nature of time. We proved a finite model property and projective unification for this logic.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42874072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-09DOI: 10.18778/0138-0680.2022.17
T. Sayed Ahmed
Let (2
让(2
{"title":"Complete Representations and Neat Embeddings","authors":"T. Sayed Ahmed","doi":"10.18778/0138-0680.2022.17","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.17","url":null,"abstract":"Let (2<n<omega). Then ({sf CA}_n) denotes the class of cylindric algebras of dimension (n), ({sf RCA}_n) denotes the class of representable (sf CA_n)s, ({sf CRCA}_n) denotes the class of completely representable ({sf CA}_n)s, and ({sf Nr}_n{sf CA}_{omega}(subseteq {sf CA}_n)) denotes the class of (n)-neat reducts of ({sf CA}_{omega})s. The elementary closure of the class ({sf CRCA}_n)s ((mathbf{K_n})) and the non-elementary class ({sf At}({sf Nr}_n{sf CA}_{omega})) are characterized using two-player zero-sum games, where ({sf At}) is the operator of forming atom structures. It is shown that (mathbf{K_n}) is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of (mathbf{S_c}{sf Nr}_n{sf CA}_{omega}) where (mathbf{S_c}) is the operation of forming complete subalgebras. For any class (mathbf{L}) such that ({sf At}{sf Nr}_n{sf CA}_{omega}subseteq mathbf{L}subseteq {sf At}mathbf{K_n}), it is proved that ({bf SP}mathfrak{Cm}mathbf{L}={sf RCA}_n), where ({sf Cm}) is the dual operator to (sf At); that of forming complex algebras. It is also shown that any class (mathbf{K}) between ({sf CRCA}_ncap mathbf{S_d}{sf Nr}_n{sf CA}_{omega}) and (mathbf{S_c}{sf Nr}_n{sf CA}_{n+3}) is not first order definable, where (mathbf{S_d}) is the operation of forming dense subalgebras, and that for any (2<n<m), any (lgeq n+3) any any class (mathbf{K}) (such that ({sf At}({sf Nr}_n{sf CA}_{m})cap {sf CRCA}_nsubseteq mathbf{K}subseteq {sf At}mathbf{S_c}{sf Nr}_n{sf CA}_{l}), (mathbf{K}) is not not first order definable either.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47903877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-02DOI: 10.18778/0138-0680.2022.14
W. Dziobiak, M. Schwidefsky
Abstract. The categorical dualities presented are: (first) for the category of bi-algebraic lattices that belong to the variety generated by the smallest non-modular lattice with complete (0,1)-lattice homomorphisms as morphisms, and (second) for the category of non-trivial (0,1)-lattices belonging to the same variety with (0,1)-lattice homomorphisms as morphisms. Although the two categories coincide on their finite objects, the presented dualities essentially differ mostly but not only by the fact that the duality for the second category uses topology. Using the presented dualities and some known in the literature results we prove that the Q-lattice of any non-trivial variety of (0,1)-lattices is either a 2-element chain or is uncountable and non-distributive.
{"title":"Categorical Dualities for Some two Categories of Lattices: An Extended Abstract","authors":"W. Dziobiak, M. Schwidefsky","doi":"10.18778/0138-0680.2022.14","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.14","url":null,"abstract":"Abstract. The categorical dualities presented are: (first) for the category of bi-algebraic lattices that belong to the variety generated by the smallest non-modular lattice with complete (0,1)-lattice homomorphisms as morphisms, and (second) for the category of non-trivial (0,1)-lattices belonging to the same variety with (0,1)-lattice homomorphisms as morphisms. Although the two categories coincide on their finite objects, the presented dualities essentially differ mostly but not only by the fact that the duality for the second category uses topology. Using the presented dualities and some known in the literature results we prove that the Q-lattice of any non-trivial variety of (0,1)-lattices is either a 2-element chain or is uncountable and non-distributive.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42997869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-11DOI: 10.18778/0138-0680.2022.07
Patryk Michalczenia
In the article we study the existence predicate (varepsilon) in the context of semantics for first-order modal logic. For a formula (varphi) we define (varphi^{varepsilon}) - the so called existence relativization. We point to a gap in the work of Fitting and Mendelsohn concerning the relationship between the truth of (varphi) and (varphi^{varepsilon}) in classes of varying- and constant-domain models. We introduce operations on models which allow us to fill the gap and provide a more general perspective on the issue. As a result we obtain a series of theorems describing the logical connection between the notion of truth of a formula with the existence predicate in constant-domain models and the notion of truth of a formula without the existence predicate in varying-domain models.
{"title":"First-Order Modal semantics and Existence Predicate","authors":"Patryk Michalczenia","doi":"10.18778/0138-0680.2022.07","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.07","url":null,"abstract":"In the article we study the existence predicate (varepsilon) in the context of semantics for first-order modal logic. For a formula (varphi) we define (varphi^{varepsilon}) - the so called existence relativization. We point to a gap in the work of Fitting and Mendelsohn concerning the relationship between the truth of (varphi) and (varphi^{varepsilon}) in classes of varying- and constant-domain models. We introduce operations on models which allow us to fill the gap and provide a more general perspective on the issue. As a result we obtain a series of theorems describing the logical connection between the notion of truth of a formula with the existence predicate in constant-domain models and the notion of truth of a formula without the existence predicate in varying-domain models.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47745870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-23DOI: 10.18778/0138-0680.2022.12
K. Sasaki
In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper as opposed to proper ones. Improper inference rules are more complicated than proper ones and more difficult to understand. In 2022, provided a sequent system based solely on the application of proper rules. In the present paper, on the basis of our system from 2022, we classify improper inference rules.
{"title":"A Classification of Improper Inference Rules","authors":"K. Sasaki","doi":"10.18778/0138-0680.2022.12","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.12","url":null,"abstract":"In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper as opposed to proper ones. Improper inference rules are more complicated than proper ones and more difficult to understand. In 2022, provided a sequent system based solely on the application of proper rules. In the present paper, on the basis of our system from 2022, we classify improper inference rules.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42687557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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