Rough bi-Heyting algebra and its applications on Rough bi-intuitionistic logic

A Rough semiring $(T,\mathrm{\Delta },\mathrm{\nabla })$ 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 ${(}^{\ast })$, dual pseudocomplement ${(}^{+})$, relative pseudocomplement $(\to )$ and dual relative pseudocomplement $(\leftarrow )$. In this paper, it is proved that the elements of the Rough semiring $(T,\mathrm{\Delta },\mathrm{\nabla })$ are accomplished with the pseudocomplement, relative pseudocomplement along with their duals. The definition of pseudocomplement leads to the concept of Brouwerian Rough semiring structure $(T,\mathrm{\Delta },\mathrm{\nabla }{,}^{\ast },RS(\mathrm{\varnothing }),RS(U))$ on the Rough semiring $(T,\mathrm{\Delta },\mathrm{\nabla })$. Also it is proved $(T,\mathrm{\Delta },\mathrm{\nabla },\to ,\leftarrow ,RS(\mathrm{\varnothing }),RS(U))$ 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.