Kumar and Banerjee (Some algebras and logics from quasiorder-generated covering-based approximation spaces. J Appl Non-classical Log 2024;34:248-68) characterized a subclass of quasiorder-generated covering-based approximation spaces for which the algebra of definable sets forms a Stone algebra. In this paper, we characterize those subclasses for which the definable sets form dual Stone, regular double Stone, linear Heyting, and well-connected Heyting algebras. As a consequence, discrete dualities of the aforementioned algebras are also obtained. Furthermore, we provide representation theorems of the aforementioned algebras in terms of rough sets determined by a quasiorder.

We establish a Cayley-type representation theorem for distributive lattices by constructing an embedding into a suitable endomorphism structure.

Rough set theory has been extensively studied in regard to its lattice structure. However, this article concerns with the (commutative) ring structure of rough set theory. We show that a finite approximation space can be identified by a cube free natural number n by providing an isomorphism between lattice of rough sets and lattice of ideals of the ring ℤ_n . We introduce a ring structure on the rough sets via the ring structure on ideals of ℤ_n . Moreover, we also classify all the rings which are isomorphic to the rings formed by the rough sets.

In the literature many generalizations of Boolean algebras exist viz. Ockham algebras, De Morgan algebras, p-algebras, Heyting algebras etc. There has been investigations into algebras in which two or more of such negations occur simultaneously. This paper investigates the class of algebras called quad algebras which encompasses both the Boolean and De Morgan algebras. Due to the presence of Boolean negation such algebras naturally possess a ring structure. In fact these algebras turn out to be equivalent with the class of rings where every element satisfies the polynomial equation x^4=x . On the other aspect, we provide a 4-valued semantics of the logic for quad algebras. The propositional logic ℒ_𝒬𝒜 of a quad algebras is shown to be sound and complete with respect to a 4-valued semantics.

Negations as modal operators have been well investigated in the literature. There are various ways of providing semantics of negations in non-classical logic. Dunn has introduced semantics of negation in compatibility frames. In this work, we study the categories of compatibility frames and $K_{i}$ algebras—the algebraic semantics of minimal logic $K_{i}$. We establish categorical duality between $K_{i}$ algebras and descriptive general compatibility frames. Thereafter we introduce frames dual to De Morgan algebras and pseudocomplemented lattices in our framework. Correspondence of these frames with the existing frames in the literature, which are dual to these algebras, is also investigated. It is further extended to dualities between various subcategories of $K_{i}$ algebras and corresponding frames. Finally, it is shown that the duality generalizes to Stone’s duality in the case of ortho frames.

This paper investigates the logical structure of the 4-element chain considered as a double Stone algebra. It has been shown that any element of a double Stone algebra can be identified as monotone ordered triplet of sets. As a consequence, we obtain the 4-valued semantics for the logic GD of double Stone algebras. Furthermore, the rough set semantics of the logic GD is provided by dividing the boundary region (uncertainty) into two disjoint subregions.

In A. Kumar, & M. Banerjee [(2012). Definable and rough sets in covering-based approximation spaces. In T. Li. (eds.), Rough sets and knowledge technology (pp. 488–495). Springer-Verlag], A. Kumar, & M. Banerjee [(2015). Algebras of definable and rough sets in quasi order-based approximation spaces. Fundamenta Informaticae, 141(1), 37–55], authors proposed a pair of lower and upper approximation operators based on granules generated by quasiorders. This work is an extension of algebraic results presented therein. A characterisation has been presented for those quasiorder-generated covering-based approximation spaces whose corresponding collections of definable and rough sets form Stone algebras. The notion of rough lattice was proposed in A. Kumar, & M. Banerjee [(2015). Algebras of definable and rough sets in quasi order-based approximation spaces. Fundamenta Informaticae, 141(1), 37–55], A. Kumar [(2020). A Study of Algebras and Logics of Rough Sets Based on Classical and Generalized Approximation Spaces. In Transactions on Rough Sets XXII, LNCS (Vol. 12485, pp. 123–251). Springer]. Some special rough lattices are introduced in this work, viz. rough Stone algebra, ∼1-complemented and ∼2-complemented rough lattices. Representations of these algebras in terms of rough sets are obtained. Moreover, logics for these algebras are shown to be sound and complete with respect to rough set semantics.

This paper explores the topological aspects of algebras determined by rough sets. It is well known that the lattice structure of rough sets is characterized by the lattices of the form 2(I) x 3(J). We show that algebras determined by the rough sets are characterized by the condition "any chain of completely prime filters has at most two elements". We introduce the notion of rough topological space in which the basic open set has at most two elements. We further provide representations of rough topological spaces in terms of topologies determined by algebras of rough sets.

In this article, we propose 3-valued semantics of the logics compatible with Stone and dual Stone algebras. We show that these logics can be considered as 3-valued by establishing soundness and completeness results. We also establish rough set semantics of these logics where the third value can be interpreted as not certain but possible.

The seminal work of Z. Pawlak [60] on rough set theory has attracted the attention of researchers from various disciplines. Algebraists introduced some new algebraic structures and represented some old existing algebraic structures in terms of algebras formed by rough sets. In Logic, the rough set theory serves the models of several logics. This paper is an amalgamation of algebras and logics of rough set theory. We prove a structural theorem for Kleene algebras, showing that an element of a Kleene algebra can be looked upon as a rough set in some appropriate approximation space. The proposed propositional logic $$\mathcal {L}_{K}$$ of Kleene algebras is sound and complete with respect to a 3-valued and a rough set semantics. This article also investigates some negation operators in classical rough set theory, using Dunn's approach. We investigate the semantics of the Stone negation in perp frames, that of dual Stone negation in exhaustive frames, and that of Stone and dual Stone negations with the regularity property in $$K_{-}$$ frames. The study leads to new semantics for the logics corresponding to the classes of Stone algebras, dual Stone algebras, and regular double Stone algebras. As the perp semantics provides a Kripke type semantics for logics with negations, exploiting this feature, we obtain duality results for several classes of algebras and corresponding frames. In another part of this article, we propose a granule-based generalization of rough set theory. We obtain representations of distributive lattices (with operators) and Heyting algebras (with operators). Moreover, various negations appear from this generalized rough set theory and achieved new positions in Dunn's Kite of negations.

Incidence of Complete Heart block is rare during pregnancy and in most cases, it is first presentation of congenital complete heart block. Acquired cases of CHB may present in women with pre-existing conditions like congenital heart disease, cardiomyopathy, systemic lupus erythematosus or associated acute/ sub-acute condition like acute myocardial infarction, metabolic disturbances, drug intoxication or acute infection. Previous reported cases presented during pregnancy, while the two cases described here presented in puerperium after uneventful delivery. One of the cases required permanent pacemaker implantation while other was managed with temporary pacemaker support and discharged in stable condition after spontaneous reversion to normal sinus rhythm.

Moisil in 1941, while constructing the algebraic models of n-valued Łukasiewicz logic defined the set $B^{[n]}$,where $B$ is a Boolean algebra and `n' being a natural number. Further it was proved by Moisil himself the representations of n-valued Łukasiewicz Moisil algebra in terms of $B^{[n]}$. In this article, structural representation results for Stone, dual Stone and double Stone algebras are proved similar to Moisil's work by showing that elements of these algebras can be looked upon as monotone ordered tuple of sets. 3-valued semantics of logic for Stone algebra, dual Stone algebras and 4-valued semantics of logic for double Stone algebras are proposed and established soundness and completeness results.

A structural theorem for Kleene algebras is proved, showing that an element of a Kleene algebra can be looked upon as an ordered pair of sets. Further, we show that negation with the Kleene property (called the `Kleene negation') always arises from the set theoretic complement. The corresponding propositional logic is then studied through a 3-valued and rough set semantics. It is also established that Kleene negation can be considered as a modal operator, and enables giving a perp semantics to the logic. One concludes with the observation that all the semantics for this logic are equivalent.

We numerically examine the effect of the cross-sectional shape on the propagation characteristics of the fundamental surface plasmon polariton (SPP) mode guided along Au and Ag nanowires. It is observed that as the cross-sectional symmetry decreases from circular to triangular via pentagonal and square, the modal confinement, loss and dispersion increases. Further the difference in propagation characteristics between circular and other cross-sections increases rapidly as the wavelength of operation decreases. The study should be useful for selecting the cross-sectional shape of the nanowire for a particular application.

This article investigates whether a few well-known 'negation' operators may be termed as negations, using Dunn's approach. The semantics of the Stone negation is investigated in perp frames, that of dual Stone negation in exhaustive frames, and that of Stone and dual Stone negations with the regularity property, in $$K_{-}$$ frames. The study leads to new semantics for the logics corresponding to the classes of Stone algebras, dual Stone algebras and regular double Stone algebras.

30Apr 2016 Improved Face Recognition Method usingPCA. Ramesh Kumar Verma , Arun Kumar Deepak Kumar and Lucknesh Kumar. Department of CSE. Assistant Professor, Department of CSE.

A pair of approximation operators, based on the notion of granules in generalized approximation spaces, was studied in an earlier work by the authors. In this article, we investigate algebraic structures formed by the definable sets and also by the rough sets determined by this pair of approximation operators. The definable sets are open sets of an Alexandrov topological space, and form a completely distributive lattice in which the set of completely join irreducible elements is join dense. The collection of rough sets also forms a similar structure. Representation results for such classes of completely distributive lattices as well as Heyting algebras in terms of definable and rough sets are obtained. Further, two unary operators on rough sets are considered, making the latter constitute a structure that is named a 'rough lattice'. Representation results for rough lattices are proved.

In this paper, localization of the robot is achieved by considering two Global Positioning Systems (GPS) or DGPS.Differential Global Positioning System (DGPS) is interfaced with MBED with the help of Zigbee protocol.For accurate localization of mobile robot DGPS is preferred.Filters are used to remove the erroneous noise from the data obtained from GPS.Low pass IIR filter for DGPS is realized.The project work discusses each of these approaches for localization in Outdoor environment.The above algorithm is implemented on MBED Platform.Simulation results are extracted using Matlab.

The study was conducted to assessthe relationship between socio-economic indicators and women empowerment in Harahua block of district Varanasi, India during 2012. One hundred twenty five rural women were sampled for the study. The relevant variables were selected based upon the available literature and the works done in this field prior to the present investigation. Age, caste, marital status, family size, family type, land holding capacity, family income, housing pattern and occupation were used as a socio-economic indicators. A structured interview schedule was used for collection of the data through personal interview method. Data was analyzed using frequency, percentage, mean, standard deviation, and chi-square test. The socio-economic profile of rural women indicated that majority of them were middle aged (51.2%), married (56%) and belonged to other backword (OBC) caste (60%). Majority belonged to small and nuclear family (63.2%) having marginal land holding (51.2); most of the respondents belong to medium size family (59.2%) i.e. 5 to 10 members under medium income ( 21900 to 34700) category (44%) having Pucca house (61.6%). Among nine selected socio-economic indicators i.e., age, marital status, caste, family type, land holding, family size, income of the family, house and occupation, only four variables have relationship with women empowerment.

This paper presents findings of an extensive survey of Indian manufacturing industries. The survey enc ompassed three sectors: Four Wheeler Automobile Industries, Two Wheeler Aut omobile Industries and General Manufacturing Indust ries. Various Improvement Activities such as Advance Manufacturin g Technology, Integrated Information System and Adv ance Management System have been identified and assessed the degree of investment in Improvement Activities in Indian manufacturing environment. Their sub classifications are also mad e. Sector wise comparisons of these Improvement Act ivities are provided. Correlation is also made between the various Improv ement Activities. Our results showed that most of t he Indian industries are still emphasizing Advanced Management Systems. Four Wheeler Automobile Industries and General Manufact uring Industries are emphasizing on Total Quality Management. Howeve r Two Wheeler Automobile Industries are investing m ore on Activity Based Costing. But, in overall sector it is observe d from data analysis that all sectors are emphasizi ng on Total Quality Management.