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We redefine a system of varieties definable by a schema of equations to include finite dimensions. Then we present a technique using ultraproducts enabling one to lift results proved for every finite dimension to the transfinite. Let Ord denote the class of all ordinals. Let ⟨Kα : α ∈ Ord⟩ be a system of varieties definable by a schema. Given any ordinal α, we define an operator Nrα that acts on Kβ for any β > α giving an algebra in Kα, as an abstraction of taking α-neat reducts for cylindric algebras. We show that for any positive k, and any infinite ordinal α that SNrαKα+k+1 cannot be axiomatized by a finite schema over SNrαKα+k given that the result is valid for all finite dimensions greater than some fixed finite ordinal. We apply our results to cylindric algebras and Halmos quasipolyadic algebras with equality. As an application to our algebraic result we obtain a strong incompleteness theorem (in the sense that validitities are not captured by finitary Hilbert style axiomatizations) for an algebraizable extension of Lω,ω. [ABSTRACT FROM AUTHOR]

Due to increasing concern about plastic waste and its impact on the ecosystem, it is vital to understand tourists' behavioral intentions about plastic waste reduction on beaches. There have been several studies that have used the theory of planned behavior to investigate pro-environmental behaviors or intentions, but there are few specific research studies that have extended the theory of planned behavior by adding awareness of consequences to explain the power of behavioral intention. Accordingly, this paper aimed to investigate how awareness of consequences, subjective norms, attitudes, and behavioral control dimensions influence plastic waste behavioral intention on Jeddah's beaches, in Saudi Arabia. This was performed on a random sample of 390 local tourists in Jeddah city from June to August 2023. A total of 340 of them agreed and answered the questionnaire, yielding a percentage response rate of 87%. This produced 271 valid questionnaires for data analysis after closely examining the survey. A self-complete questionnaire was used for data collection in using multiple statistical analyses to examine the hypotheses. The results demonstrated a positive influence of subjective norms, perceived behavioral control, and consequence awareness on environmental behavioral intention. On the other hand, attitude did not significantly contribute to predict environmental behavioral intention. This study's findings made clear how crucial it is to consider any potential negative effects while making plans to cut down on plastic trash. Educating people about the possible harm that using plastic products on beaches is anticipated to cause to the environment might also be beneficial. It is intended that, through understanding behavior and behavior determinants, governmental bodies, pro-environmental organizations, businesses, and communities will be able to implement appropriate strategies to reduce the use of plastic in Saudi Arabia to protect marine life. [ABSTRACT FROM AUTHOR]

Let 2 < n < ω. Then CAn denotes the class of cylindric algebras of dimension n, RCAn denotes the class of representable CAns, CRCAn denotes the class of completely representable CAns, and NrnCAω(⊆ CAn) denotes the class of n-neat reducts of CAωs. The elementary closure of the class CRCAns (Kn) and the nonelementary class At(NrnCAω) are characterized using two-player zero-sum games, where At is the operator of forming atom structures. It is shown that Kn is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of ScNrnCAω where Sc is the operation of forming complete subalgebras. For any class L such that AtNrnCAω ⊆ L ⊆ AtKn, it is proved that SPCmL = RCAn, where Cm is the dual operator to At; that of forming complex algebras. It is also shown that any class K between CRCAn ∩ SdNrnCAω and ScNrnCAn+3 is not first order definable, where Sd is the operation of forming dense subalgebras, and that for any 2 < n < m, any l ≥ n + 3 any any class K such that At(NrnCAm ∩ CRCAn) ⊆ K ⊆ AtScNrnCAl, K is not not first order definable either. [ABSTRACT FROM AUTHOR]

Fix a finite ordinal n ≥ 3 and let α be an arbitrary ordinal. Let CAn denote the class of cylindric algebras of dimension n and RA denote the class of relation algebras. Let PAα(PEAα) stand for the class of polyadic (equality) algebras of dimension α. We reprove that the class CRCAn of completely representable CAns, and the class CRRA of completely representable RAs are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety V between polyadic algebras of dimension n and diagonal free CAns. We show that that the class of completely and strongly representable algebras in V is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class CRRA is not closed under =1,ω. In contrast, we show that given α ≥ ω, and an atomic A 2 PEAα, then for any n < !, NrnA is a completely representable PEAn. We show that for any α ≥ ω, the class of completely representable algebras in certain reducts of PAαs, that happen to be varieties, is elementary. We show that for α ≥ ω, the the class of polyadic-cylindric algebras dimension α, introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension n and PAαs for α an infinite ordinal, proving negative results for the first and positive ones for the second. [ABSTRACT FROM AUTHOR]

Let α be an ordinal and L be a unimodal logic (like S4 or S5). A modal cylindric algebra of dimension α, an LCAα, is a cylindric algebra of dimension α, expanded with α-many L modalities. For a frame (U, R) of L, each k < α, one defines a diamond box operator on. This defines the semantics of the L modalities in set algebras, with the rest of the operations defined like in cylindric set algebras of dimension α. We study interpolation properties for the corresponding predicate logic having α-many variables. Our results are valid for any reflexive L whose frames contain the universal frames (U, U × U). In particular, they hold for K5CAα, S4CAα (which is an algebraizable extension of topological predicate logic with semantics induced by Alexandrov topologies). [ABSTRACT FROM AUTHOR]

Fix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of Ln. Members of RCAn are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever 𝔄 ∈ V is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated Henkin-Orey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of Lω,ω, fails dramatically for Ln even if we allow certain generalized models that are only locallly clasfsical. It is also shown that any class K such that N r n C A ω ∩ CRC A n ⊆ ¯ K ⊆ ¯ S c N r n C A n + 3 , where CRCAn is the class of completely representable CAns, and Sc denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that S d RaC A ω ⊆ ¯ K ⊆ ¯ S c RaC A 5 is not elementary, where Sd denotes the operation of forming dense subalgebra. [ABSTRACT FROM AUTHOR]

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability. [ABSTRACT FROM AUTHOR]

For β an ordinal, let PEAβ (SetPEAβ) denote the class of polyadic equality (set) algebras of dimension β. We show that for any infinite ordinal α, if A ∈ PEA α is atomic, then for any n < ω, the n-neat reduct of A , in symbols ℜ r n A → B , is a completely representable PEAn (regardless of the representability of A). That is to say, for all non-zero a ∈ ℜ r n A , there is a B a ∈ SetPEA n and a homomorphism f a : ℜ r n A → B such that fa(a) ≠ 0 and f a (∑ X) = ∪ x ∈ X f a (x) for any X ⊂ = A for which ∑ X exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under ≡∞,ω. Various notions of representability (such as 'satisfying the Lyndon conditions', weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CAns satisfying the Lyndon conditions coincides with the class of atomic algebras in ElScNrnCAω, where El denotes 'elementary closure' and Sc is the operation of forming complete subalgebras. [ABSTRACT FROM AUTHOR]

This study aims to investigate the impact of the Hail International Rally event on the host community of Hail. A survey was conducted among a random sample of Hail local residents which was selected in a convenient manner from various public places. The data collection was conducted starting from the last week of March 2016 (the period of organizing the event and continued till after the end of a month. Out of 250 questionnaires that were distributed, only 137 ones were returned, yielding 127 completed and usable packets. The results showed that there was a significant correlation between the level of support and the positive impact perceived regarding urban facilities, economic benefits except the contribution of the event to achieve the commercial boom for small businesses, image, cultural communication, entertainment, community participation, and social relations. [ABSTRACT FROM AUTHOR]

We prove several interpolation theorems for many-valued infinitary logic with quantifiers by studying expansions of MV-algebras in the spirit of polyadic and cylindric algebras. We prove for various reducts of polyadic MV-algebras of infinite dimensions that ifis the free algebra in the given signature,,is in the subalgebra ofgenerated by,is in the subalgebra ofgenerated byand, then there exists an interpolantin the subalgebra generated byandsuch that. We call this a varying interpolation property because the integerdepends on the inequality. We also address cases where this interpolation property fails, but other weaker (also varying) ones hold. One such interpolation theorem says that though an interpolantmay not be found as above, an interpolant can always be found if finitely many universal quantifiers are applied tomaking it smaller and the same number of existential quantifiers are applied tomaking it bigger. This number of quantifiers also varies; it depends on the inequality. Several amalgamation theorems for classes (mostly varieties) of polyadic MV-algebras are obtained. Completeness theorems, relative to Hilbert-style axiomatisations, for the corresponding infinitary many-valued predicate logics are derived using the methodology of algebraic logic. [ABSTRACT FROM PUBLISHER]

This research attempts to investigate the impact of information sources on the decision making process of travel to the Egyptian tourist destination after the revolution of January 25, 2011. A questionnaire was downloaded on the Website of (monkey survey) during the period (12 October -11December 2012). It has been posted at many tourism websites (tourism forums, many tourism organizations (international, regional and domestic), besides sending about 2000 e-messages randomly using the facebook website and LinkedIn website. The results revealed that the least trusted sources of information about Egypt after the revolution of January 25, 2011 are the Egyptian mass media. Gender and region have significant effect on the reaction of respondents about the decision making process of travel to Egypt after the revolution. The marital status has a significant effect on respondents' trust in the Egyptian mass media. There is a statistically difference between respondents wiThrespect to their region, gender, marital status, and income regarding their perception about the role which mass media could play in their decision of travel to Egypt. Significant contribution of this paper is the conceptual model that explains the impact of information sources, particularly mass media on the decision making process of travel to Egypt after the revolution of January 25, 2011. From 133 answered questionnaires which were distributed, only 101 were valid. Data extracted have been tabulated and analyzed using SPSS 19.0 statistical package [ABSTRACT FROM AUTHOR]

Using games, as introduced by Hirsch and Hodkinson in algebraic logic, we give a recursive axiomatization of the class of representable quasi-polyadic equality algebras of any dimension α. Following Sain and Thompson in modifying Andréka's methods of splitting, to adapt the quasi-polyadic equality case, we show that if Σ is a set of equations axiomatizing for and , k′ < ω are natural numbers, then Σ contains infinitely equations in which − occurs, one of + or · occurs, a diagonal or a permutation with index l occurs, more than k cylindrifications and more than k′ variables occur. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim [ABSTRACT FROM AUTHOR]

We show that several varieties of algebras studied in algebraic logic fail to have the amalgamation property. [ABSTRACT FROM PUBLISHER]

Let 1 < n < m ≤ ω. We investigate the following question: For which reducts of CAm is the class of neat n - reducts (not) elementary. We also characterize the class of neat reducts using games. [ABSTRACT FROM AUTHOR]

We show that for infinite ordinals α the class of polyadic algebras of dimension α has the super amalgamation property (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the (seemingly very innocent) fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new unpublished results (answering long-standing open problems in the field) concerning neat reducts and amalgamation are given. (Theorems 11, 13, 19 and 31-38 are such). Several counterexamples which convey the gist of techniques used in this area are presented two of which are new (Theorem 19, Theorem 38.) It is known that the algebraic notion of amalgamation in a class of algebras corresponds to the metalogical notion of interpolation in the corresponding logic. Answers to open question in the recent paper [31] concerning both amalgamation and interpolation are summarized in tabular form at the end of this paper. [ABSTRACT FROM PUBLISHER]

Introduced by Leon Henkin back in the fifties, the notion of neat reducts is an old venerable notion in algebraic logic. But it is often the case that an unexpected viewpoint yields new insights. Indeed, the repercussions of the (seemingly very innocent) fact that the class of neat reducts is not closed under forming subalgebras turn out to be enormous. In this paper we review and, in the process, discuss, some of these repercussions in connection with the algebraic notion of amalgamation. Some new unpublished results (answering long-standing open problems in the field) concerning neat reducts and amalgamation are given. (Theorems 11, 13, 19 and 31-38 are such). Several counterexamples which convey the gist of techniques used in this area are presented two of which are new (Theorem 19, Theorem 38). It is known that the algebraic notion of amalgamation in a class of algebras corresponds to the metalogical notion of interpolation in the corresponding logic. Answers to open question in the recent paper [54] concerning both amalgamation and interpolation are summarized in tabular form at the end of this paper. [ABSTRACT FROM PUBLISHER]

We give a simple new construction of representable relation algebras with non-representable completions. Using variations on our construction, we show that the elementary closure of the class of completely representable relation algebras is not finitely axiomatizable (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

We show that it is impossible to define a substitution operator for arbitrary representable cylindric algebras that agrees in its basic properties with the notion of substitutions introduced for dimension complemented algebras (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Letn > 2. A weakly representable relation algebra that is not strongly representable is constructed. It is proved that the set of all n by n basic matrices forms a cylindric basis that is also a weakly but not a strongly representable atom structure. This gives an example of a binary generated atomic representable cylindric algebra with no complete representation. An application to first order logic is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

In this paper, an extension of first order logic is introduced. In such logics atomic formulas may have infinite lengths. An Omitting Types Theorem is proved. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

We solve some open questions on the amalgamation porperty in various algebras of logic (like cylindric algebras). Some of these problems first appeared in Pigozzi [3]. [ABSTRACT FROM PUBLISHER]

SC, CA, QA and QEA denote the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasi-polyadic algebras and quasi-polyadic equality algebras, respectively. Let ω ≤ α < β and let K ∈ {SC,CA,QA,QEA}. We show that the class of α -dimensional neat reducts of algebras in Kβ is not elementary. This solves a problem in [3]. Also our result generalizes results proved in [2] and [3]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Let 1 < n < ω and β > n. We show that the class NrnCAβ of n-dimensional neat reducts of β-dimensional cylindric algebras is not closed under forming elementary subalgebras. This solves a long-standing open problem of Tarski and his co-authors Andréka, Henkin, Monk and Németi. The proof uses genuine model-theoretic arguments. [ABSTRACT FROM AUTHOR]

We show that certain properties of dimension complemented cylindric algebras, concerning neat embeddings, do not generalize much further. Let α ≥ ω. There are non-isomorphic representable cylindric algebras of dimension α each of which is a generating subreduct of the same β dimensional cylindric algebra. We also show that there exists a representable cylindric algebra &Afr; of dimension α, such that &Afr; is a generating subreduct of &Bfr; and &Bfr;′, both in CAα +ω, however &Bfr; and &Bfr;′ are not isomorphic. This settle questions raised by Henkin, Monk and Tarski (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]