Fuzzy Hyper and Pythagorean Fuzzy Composite Structures on BCL–Algebra and LiuB–Algebra

Abebe, Asmamaw

dc.contributor.author	Abebe, Asmamaw
dc.date.accessioned	2026-06-05T07:50:05Z
dc.date.available	2026-06-05T07:50:05Z
dc.date.issued	2025-11
dc.identifier.uri	http://ir.bdu.edu.et/handle/123456789/16867
dc.description.abstract	This dissertation presents det ailed and thorough research on the notions of hyper BCL–algebra, fuzzy substructures in hyper BCL–algebra, fuzzy substructures in BCL–algebra, Pythagorean fuzzy substructures in BCL–algebra, fuzzy substructures in LiuB–algebra and Pythagorean fuzzy substructures in LiuB–algebra. It provides new theoretical understanding, well refined definitions and rigorous characterizations that contribute to the development of algebraic knowledge about these structures. The research begins by introducing the hyper BCL–algebraic structures, supported by proven properties, under which (weak, strong) hyper subalgebras, (weak, strong) hyper deductive systems and (weak, strong) hyper ideals of hyper BCL–algebra are defined and several relevant properties are investigated. The relations among strong hyper subalgebras, weak hyper subalgebras and hyper subalgebras, as well as among strong hyper ideals, weak hyper ideals and hyper ideals of hyper BCL– algebra are clearly demonstrated in the context of hyper BCL–algebras. In hyper BCL–algebras, the relationship between hyper deductive systems and weak deductive systems is established. The intersection and union of corresponding (weak, strong) hyper substructures of hyper BCL–algebras are shown to be conserved. Following the introduction of hyper substructures, the notions of fuzzy hyper algebras are introduced, characterized as fuzzy (weak, strong) hyper subalgebras, fuzzy (weak, strong) hyper deductive systems and fuzzy (weak, strong) hyper ideals. The conservation of intersections of corresponding substructures is established; however, unions of such substructures are generally not conserved justified by examples. The relationships among fuzzy (weak, strong) hyper substructures are also explored. After investigating additional relevant properties, we introduce the notions of fuzzy subalgebra, fuzzy deductive system and fuzzy ideal of BCL–algebra. Moreover, we prove that the complements of fuzzy substructures of BCL–algebra and its characteristic functions correspond to fuzzy substructures of BCL–algebra. In the BCL–algebra, we also prove that intersections of fuzzy subalgebras, fuzzy deductive systems and fuzzy ideals are fuzzy subalgebra, fuzzy deductive system and fuzzy ideal of BCL–algebra, respectively; however, unions of such substructures are not conserved justified by counter examples. Several additional properties of fuzzy substructures of BCL–algebra are also demonstrated. Following the introduction and investigation of fuzzy subsets of BCL–algebra, we extend these notions to Pythagorean fuzzy substructures of BCL–algebra; viii namely, Pythagorean fuzzy subalgebra, Pythagorean fuzzy deductive system and Pythagorean fuzzy ideal of BCL–algebra along with an investigation of their relevant properties. After introducing the Pythagorean fuzzy substructures of BCL–algebra, we present new definitions of LiuB–algebra based on BCL–algebra combined with a semi–group. These definitions are illustrated with examples and their properties are explored. Following these investigations, we introduce fuzzy substructures of LiuB–algebras, including fuzzy subalgebras, fuzzy deductive systems and fuzzy ideals. In relation to substructures of LiuB–algebra, we extend these notions to Pythagorean fuzzy subalgebra, Pythagorean fuzzy deductive system and Pythagorean fuzzy ideal and relevant results are derived, with their properties explored in detail. Finally, the fuzzy subsets of BCL-algebra and LiuB–algebra and Pythagorean fuzzy sets of BCL-algebra and LiuB–algebra are described by making use of some basic tools. These include the Cartesian products of fuzzy subalgebras in BCL-algebra, the level sets in Pythagorean fuzzy deductive systems of BCL-algebra, and the homomorphisms of Pythagorean fuzzy deductive systems of LiuB–algebra. In addition, LiuB–algebra is also described through the Pythagorean ( , )–fuzzy ideal, and related properties arising from each of these descriptions are carefully examined. ix	en_US
dc.language.iso	en_US	en_US
dc.subject	Mathematics	en_US
dc.title	Fuzzy Hyper and Pythagorean Fuzzy Composite Structures on BCL–Algebra and LiuB–Algebra	en_US