Shift Filter of Quasi-ordered Residuated Systems
Abstract
The concept of residuated relational systems ordered under a quasi-order relation was introduced in 2018 by S. Bonzio and I. Chajda as a structure $\mathfrak{A} = \langle A, \cdot, \rightarrow, 1, \preccurlyeq \rangle$, where $(A,\cdot)$ is a commutative semigroup with the identity $1$ as the top element in this ordered monoid under a quasi-order $\preccurlyeq$. In 2020, the author introduced and analyzed the concepts of filters in this type of algebraic structures. In addition to the previous, the author continued to investigate some of the types of filters in quasi-ordered residuated systems such as, for example, implicative and comparative filters. In this article, as a continuation of previous author's research, the author introduced and analyzed the concepts of shift filters of quasi-ordered residuated systems and then compared it with other types of filters.
Keywords
Quasi-ordered residuated system, filter, implicative filter, comparative filter, shift filter
References
- [1] S. Bonzio, Algebraic structures from quantum and fuzzy logics, Ph.D Thesis, Cagliari: Universit‘a degli studi di Cagliari, 2015.
- [2] S. Bonzio, I. Chajda, Residuated relational systems, Asian-European J. Math., 11(2) (2018), 1850024
- [3] D. A. Romano, Filters in residuated relational system ordered under quasi-order, Bull. Int. Math. Virtual Inst., 10(3) (2020), 529-534.
- [4] D. A. Romano, Implicative filters in quasi-ordered residuated dystem, Proyecciones J. Math., 40(2) (2021), 417-424.
- [5] D. A. Romano, Associated filters in quasi-ordered residuated systems, Contributions to Mathematics, 1 (2020), 22-26.
- [6] D. A. Romano, Comparative filter in quasi-ordered residuated sysyem, Bull. Int. Math. Virtual Inst., 11(1) (2021), 177-184.
- [7] D. A. Romano, Ideals in quasi-ordered residuated system, Contrib. Math., 3 (2021), 68-76.
- [8] D. A. Romano, Strong quasi-ordered residuated system, Open J. Math. Sci., 5 (2021), 73-79.
- [9] D. A. Romano, Prime and irreducible filters in strong quasi-ordered residuated systems, Open J. Math. Sci., 5 (2021), 172-181.
- [10] D. A. Romano, Three types of prime filters in strong quasi-ordered residuated systems, An. Univ. Craiova, Ser. Mat. Inf., 49(1) (2022), 135-145.
