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1

Completion in a common supercategory of Met, UAP, wsAP and Near

Khare M., Tiwari S.

Demonstratio Mathematica

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2013

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Vol. 46, nr 1

209--227

EN

This paper investigates the notion of approach nearness spaces. Using clusters, completion of an approach nearness space is constructed, which is a unified study of completion in the context of metric spaces, uniform approach spaces, weakly symmetric approach spaces and nearness spaces. Another generalization of completeness, called ultrafilter completeness is introduced to prove the Niemytzki–Tychonoff theorem for approach nearness spaces. Both definitions of completions are shown to be equivalent in a limit-regular approach space. Various examples are given to support the present study.

2

L-approach merotopies and their categorical perspective

Khare M., Tiwari S.

Demonstratio Mathematica

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2012

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Vol. 45, nr 3

699-716

EN

In the present paper, we have made a category theoretic and lattice theoretic study of some nearness-like structures in the L-approach theory. Using L-grills, the notion of L-approach grill structure is introduced as a characterization of L-approach grill merotopy on X; their categorical perspectives and implications are also investigated. A number of illustrative examples are included.

3

Grill Determined L-Approach Merotopological Spaces

Khare M., Tiwari S.

Fundamenta Informaticae

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2010

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Vol. 99, nr 1

1-12

EN

The present paper is devoted to the study of grill determined L-approachmerotopological spaces. The category having such spaces as objects is shown to be a topological construct (its initial and final structures are provided explicitly). The lattice structure of the family of all these spaces is also discussed. In the classical theory, this category (that is, when L = {0, 1}) is a supercategory of the category of pseudo metric spaces and nonexpansive maps.

4

The range of non-atomic measures on effect algebras

Khare M., Singh A. K.

Demonstratio Mathematica

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2010

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Vol. 43, nr 3

497-510

EN

The present paper deals with the study of superior variation m+, inferior variation m¯ and total variation |m| of an extended real-valued function m defined on an effect algebra L; having obtained a Jordan type decomposition theorem for a locally bounded real-valued measure m defined on L, we have observed that the range of a non-atomic function m defined on a D-lattice L is an interval (—m¯ (1), m+(1)). Finally, after introducing the notion of a relatively non-atomic measure on an effect algebra L, we have proved an analogue of Lyapunov convexity theorem for this measure.

5

Valuation and metric on a lattice effect algebra

Khare M., Gupta S.

Demonstratio Mathematica

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2008

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Vol. 41, nr 2

249-258

EN

In the present paper we have derived some properties of the pseudo-metric introduced by Riećanova on a lattice effect algebra E corresponding to a valuation u on E, which turns out to be a metric if u happens to be faithful. Using these properties we have been able to prove that this metric is complete. Also it is observed that the resulting metric space is convex if and only if w is non-atomic if and only if E is atomless.

6

Relations between various continuity concepts in effect algebras

Khare M., Gupta S.

Demonstratio Mathematica

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2008

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Vol. 41, nr 3

507-518

EN

The aim of the present paper is to establish relations between continuity concepts for a nonnegative extended real-valued function [...] defined on an effect algebra. Examples and counterexamples are given to illustrate various situations arising in this study.