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Keywords = MoBiNad set

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14 pages, 267 KiB  
Article
Introduction to Non-Standard Neutrosophic Topology
by Mohammed A. Al Shumrani and Florentin Smarandache
Symmetry 2019, 11(5), 706; https://doi.org/10.3390/sym11050706 - 23 May 2019
Viewed by 3039
Abstract
For the first time we introduce non-standard neutrosophic topology on the extended non-standard analysis space, called non-standard real monad space, which is closed under neutrosophic non-standard infimum and supremum. Many classical topological concepts are extended to the non-standard neutrosophic topology, several theorems and [...] Read more.
For the first time we introduce non-standard neutrosophic topology on the extended non-standard analysis space, called non-standard real monad space, which is closed under neutrosophic non-standard infimum and supremum. Many classical topological concepts are extended to the non-standard neutrosophic topology, several theorems and properties about them are proven, and many examples are presented. Full article
25 pages, 300 KiB  
Article
Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis
by Florentin Smarandache
Symmetry 2019, 11(4), 515; https://doi.org/10.3390/sym11040515 - 10 Apr 2019
Cited by 5 | Viewed by 2282
Abstract
We extend for the second time the nonstandard analysis by adding the left monad closed to the right, and right monad closed to the left, while besides the pierced binad (we introduced in 1998) we add now the unpierced binad—all these in order [...] Read more.
We extend for the second time the nonstandard analysis by adding the left monad closed to the right, and right monad closed to the left, while besides the pierced binad (we introduced in 1998) we add now the unpierced binad—all these in order to close the newly extended nonstandard space under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations. Then, we extend the Nonstandard Neutrosophic Logic, Nonstandard Neutrosophic Set, and Nonstandard Probability on this Extended Nonstandard Analysis space, and we prove that it is a nonstandard neutrosophic lattice of first type (endowed with a nonstandard neutrosophic partial order) as well as a nonstandard neutrosophic lattice of second type (as algebraic structure, endowed with two binary neutrosophic laws: infN and supN). Many theorems, new terms introduced, better notations for monads and binads, and examples of nonstandard neutrosophic operations are given. Full article
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