Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis
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 u...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-04-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/4/515 |
| _version_ | 1798040252326084608 |
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| author | Florentin Smarandache |
| author_facet | Florentin Smarandache |
| author_sort | Florentin Smarandache |
| collection | DOAJ |
| description | 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: <i>inf<sub>N</sub></i> and <i>sup<sub>N</sub></i>). Many theorems, new terms introduced, better notations for monads and binads, and examples of nonstandard neutrosophic operations are given. |
| first_indexed | 2024-04-11T22:04:54Z |
| format | Article |
| id | doaj.art-2c130e970bc443c98488b5e3955717fc |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T22:04:54Z |
| publishDate | 2019-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-2c130e970bc443c98488b5e3955717fc2022-12-22T04:00:44ZengMDPI AGSymmetry2073-89942019-04-0111451510.3390/sym11040515sym11040515Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard AnalysisFlorentin Smarandache0Mathematics Department, University of New Mexico, Gallup, NM 87301, USAWe 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: <i>inf<sub>N</sub></i> and <i>sup<sub>N</sub></i>). Many theorems, new terms introduced, better notations for monads and binads, and examples of nonstandard neutrosophic operations are given.https://www.mdpi.com/2073-8994/11/4/515nonstandard analysisextended nonstandard analysisopen and closed monads to the left/rightpierced and unpierced binadsMoBiNad setinfinitesimalsinfinitiesnonstandard realsstandard realsnonstandard neutrosophic lattices of first type (as poset) and second type (as algebraic structure), nonstandard neutrosophic logicextended nonstandard neutrosophic logicnonstandard arithmetic operationsnonstandard unit intervalnonstandard neutrosophic infimumnonstandard neutrosophic supremum |
| spellingShingle | Florentin Smarandache Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis Symmetry nonstandard analysis extended nonstandard analysis open and closed monads to the left/right pierced and unpierced binads MoBiNad set infinitesimals infinities nonstandard reals standard reals nonstandard neutrosophic lattices of first type (as poset) and second type (as algebraic structure), nonstandard neutrosophic logic extended nonstandard neutrosophic logic nonstandard arithmetic operations nonstandard unit interval nonstandard neutrosophic infimum nonstandard neutrosophic supremum |
| title | Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis |
| title_full | Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis |
| title_fullStr | Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis |
| title_full_unstemmed | Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis |
| title_short | Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis |
| title_sort | extended nonstandard neutrosophic logic set and probability based on extended nonstandard analysis |
| topic | nonstandard analysis extended nonstandard analysis open and closed monads to the left/right pierced and unpierced binads MoBiNad set infinitesimals infinities nonstandard reals standard reals nonstandard neutrosophic lattices of first type (as poset) and second type (as algebraic structure), nonstandard neutrosophic logic extended nonstandard neutrosophic logic nonstandard arithmetic operations nonstandard unit interval nonstandard neutrosophic infimum nonstandard neutrosophic supremum |
| url | https://www.mdpi.com/2073-8994/11/4/515 |
| work_keys_str_mv | AT florentinsmarandache extendednonstandardneutrosophiclogicsetandprobabilitybasedonextendednonstandardanalysis |