Rough bi-Heyting algebra and its applications on Rough bi-intuitionistic logic
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| Format: | Preprint |
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2022
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| _version_ | 1866914062875492352 |
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| author | Praba, B. Freeda, L. P. Anto |
| author_facet | Praba, B. Freeda, L. P. Anto |
| contents | A Rough semiring $(T,Δ,\nabla)$ is considered to describe a special distributive Rough semiring known as a Rough bi-Heyting algebra. A bi-Heyting algebra is an extension of boolean algebra and it is accomplished by weaker notion of complements namely pseudocomplement $(^{*})$, dual pseudocomplement $(^{+})$, relative pseudocomplement $(\rightarrow)$ and dual relative pseudocomplement $(\leftarrow)$. In this paper, it is proved that the elements of the Rough semiring $(T,Δ,\nabla)$ are accomplished with the pseudocomplement, relative pseudocomplement along with their duals. The definition of pseudocomplement leads to the concept of Brouwerian Rough semiring structure $(T,Δ,\nabla,^{*},RS(\emptyset),RS(U))$ on the Rough semiring $(T,Δ,\nabla)$. Also it is proved $(T,Δ,\nabla,\rightarrow,\leftarrow,RS(\emptyset),RS(U))$ is a Rough bi-Heyting algebra. The concepts are illustrated with the examples. As an application, this Rough bi-Heyting algebra is used to model Rough bi-intuitionistic logic. The syntax is defined and three types of semantics for Rough bi-intuitionistic logic are defined and validated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_11851 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rough bi-Heyting algebra and its applications on Rough bi-intuitionistic logic Praba, B. Freeda, L. P. Anto Rings and Algebras A Rough semiring $(T,Δ,\nabla)$ is considered to describe a special distributive Rough semiring known as a Rough bi-Heyting algebra. A bi-Heyting algebra is an extension of boolean algebra and it is accomplished by weaker notion of complements namely pseudocomplement $(^{*})$, dual pseudocomplement $(^{+})$, relative pseudocomplement $(\rightarrow)$ and dual relative pseudocomplement $(\leftarrow)$. In this paper, it is proved that the elements of the Rough semiring $(T,Δ,\nabla)$ are accomplished with the pseudocomplement, relative pseudocomplement along with their duals. The definition of pseudocomplement leads to the concept of Brouwerian Rough semiring structure $(T,Δ,\nabla,^{*},RS(\emptyset),RS(U))$ on the Rough semiring $(T,Δ,\nabla)$. Also it is proved $(T,Δ,\nabla,\rightarrow,\leftarrow,RS(\emptyset),RS(U))$ is a Rough bi-Heyting algebra. The concepts are illustrated with the examples. As an application, this Rough bi-Heyting algebra is used to model Rough bi-intuitionistic logic. The syntax is defined and three types of semantics for Rough bi-intuitionistic logic are defined and validated. |
| title | Rough bi-Heyting algebra and its applications on Rough bi-intuitionistic logic |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2208.11851 |