Mathematics and Statistics 10(5): 1121-1126, 2022 DOI: 10.13189/ms.2022.100523 http://www.hrpub.org Anti-hesitant Fuzzy Subalgebras, Ideals and Deductive Systems of Hilbert Algebras Aiyared Iampan 1,* , S. Yamunadevi 2 , P. Maragatha Meenakshi 3 , N. Rajesh 4 1 Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 2 Department of Mathematics, Vivekanandha College of Arts and Science for Women, Tiruchengode-637205, Tamilnadu, India 3 Department of Mathematics, Periyar E.V.R College, Tiruchirappalli (affiliated to Bharathidasan University), Tiruchirappalli 620023, Tamilnadu, India 4 Department of Mathematics, Rajah Serfoji Government College, Thanjavur 613005, Tamilnadu, India Received June 5, 2022; Revised September 13, 2022; Accepted September 21, 2022 Cite This Paper in the following Citation Styles (a): [1] Aiyared Iampan, S. Yamunadevi, P. Maragatha Meenakshi, N. Rajesh, ”Anti-hesitant Fuzzy Subalgebras, Ideals and Deductive Systems of Hilbert Algebras,” Mathematics and Statistics, Vol.10, No.5, pp. 1121-1126, 2022. DOI: 10.13189/ms.2022.100523 (b): Aiyared Iampan, S. Yamunadevi, P. Maragatha Meenakshi, N. Rajesh, (2022). Anti-hesitant Fuzzy Subalgebras, Ideals and Deductive Systems of Hilbert Algebras. Mathematics and Statistics, 10(5), 1121-1126. DOI: 10.13189/ms.2022.100523 Copyright ©2022 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract The Hilbert algebra, one of several algebraic structures, was first described by Diego in 1966 [7] and has since been extensively studied by other mathematicians. Torra [18] was the first to suggest the idea of hesitant fuzzy sets (HFSs) in 2010, which is a generalization of the fuzzy sets defined by Zadeh [20] in 1965 as a function from a reference set to a power set of the unit interval. The significance of the ideas of hesitant fuzzy subalgebras, ideals, and filters in the study of the different logical algebras aroused our interest in applying these concepts to Hilbert algebras. In this paper, the concepts of HFSs to subalgebras (SAs), ideals (IDs), and deductive systems (DSs) of Hilbert algebras are introduced in terms of anti-types. We call them anti-hesitant fuzzy subalgebras (AHFSAs), anti-hesitant fuzzy ideals (AHFIDs), and anti-hesitant fuzzy deductive systems (AHFDSs). The relationships between AHFSAs, AHFIDs, and AHFDSs and their lower and strong level subsets are provided. As a result of the study, we found their generalization as follows: every AHFID of a Hilbert algebra Ω is an AHFSA and an AHFDS of Ω. We also study and find the conditions for the complement of an HFS to be an AHFSA, an AHFID, and an AHFDS. In addition, the relationships between the complements of AHFSAs, AHFIDs, and AHFDSs and their upper and strong level subsets are also provided. Keywords Hilbert Algebra, Anti-hesitant Fuzzy Subal- gebra, Anti-hesitant Fuzzy Ideal and Anti-hesitant Fuzzy Deductive System 1 Introduction Zadeh introduced the idea of fuzzy sets (FSs) in [20]. Nu- merous academics have studied the notion of FSs, which have numerous applications in everyday life. Numerous investi- gations were undertaken into the generalizations of FSs. In [1, 2, 5], it is explained how FSs can be integrated with some uncertainty-reduction strategies like soft sets and rough sets. The concept of HFSs, a function from a reference set to a power set of the unit interval, was introduced in 2009–2010 by Torra and Narukawa [18, 19]. The idea of FSs has been expanded to include HFSs. Numerous uses for the HFS ideas created by Torra and others can be found outside of mathemat- ics and in it. Following Torra and Narukawa’s [18, 19] intro- duction of the concept of HFSs, numerous studies on its gener- alizations and applications to numerous logical algebras have been conducted. Examples of related articles can be found at [3, 4, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 22]. In this paper, the concepts of HFSs to SAs, IDs, and DSs of Hilbert algebras are introduced in terms of anti-types. The relationships between AHFSAs, AHFIDDs, and AHFDSs and their lower and strong level subsets are provided. In addition, the relationships between the complements of AHFSAs, AH-