This paper aims to study bounded algebras in another perspective-dual ideals of bounded $ L $-algebras. As the dual concept of ideals in $ L $-algebras, dual ideals are designed to characterize some significant properties of bounded $ L $-algebras. We begin by providing a definition of dual ideals and discussing the relationships between ideals and dual ideals. Then, we prove that these dual ideals induce congruence relations and quotient $ L $-algebras on bounded $ L $-algebras. Naturally, in order to construct the first isomorphism theorem between bounded $ L $-algebras, the relationship between dual ideals and morphisms between bounded $ L $-algebras is investigated and that the kernels of any morphisms between bounded $ L $-algebras are dual ideals is proven. Fortunately, although the first isomorphism theorem between arbitrary bounded $ L $-algebras fails to be proven when using dual ideals, the theorem was proven when the range of morphism was good. Another main purpose of this study is to use dual ideals to characterize several kinds of bounded $ L $-algebras. Therefore, first, the properties of dual ideals in some special bounded $ L $-algebras are studied; then, some special bounded $ L $-algebras are characterized by dual ideals. For example, a good $ L $-algebra is a $ CL $-algebra if and only if every dual ideal is $ C $ dual ideal is proven.
Citation: Chun Ge Hu, Xiao Guang Li, Xiao Long Xin. Dual ideal theory on L-algebras[J]. AIMS Mathematics, 2024, 9(1): 122-139. doi: 10.3934/math.2024008
This paper aims to study bounded algebras in another perspective-dual ideals of bounded $ L $-algebras. As the dual concept of ideals in $ L $-algebras, dual ideals are designed to characterize some significant properties of bounded $ L $-algebras. We begin by providing a definition of dual ideals and discussing the relationships between ideals and dual ideals. Then, we prove that these dual ideals induce congruence relations and quotient $ L $-algebras on bounded $ L $-algebras. Naturally, in order to construct the first isomorphism theorem between bounded $ L $-algebras, the relationship between dual ideals and morphisms between bounded $ L $-algebras is investigated and that the kernels of any morphisms between bounded $ L $-algebras are dual ideals is proven. Fortunately, although the first isomorphism theorem between arbitrary bounded $ L $-algebras fails to be proven when using dual ideals, the theorem was proven when the range of morphism was good. Another main purpose of this study is to use dual ideals to characterize several kinds of bounded $ L $-algebras. Therefore, first, the properties of dual ideals in some special bounded $ L $-algebras are studied; then, some special bounded $ L $-algebras are characterized by dual ideals. For example, a good $ L $-algebra is a $ CL $-algebra if and only if every dual ideal is $ C $ dual ideal is proven.
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Chun Ge Hu, Xiao Guang Li, Xiao Long Xin. Dual ideal theory on L-algebras[J]. AIMS Mathematics, 2024, 9(1): 122-139. doi: 10.3934/math.2024008
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