Study of Cubic Q-ideals in Q-algebras

The concept of cubic ideals is an important generalization in an algebraic structure. In this work, we introduce the notion of a cubic Q-sub-algebra and a cubic Q-ideal within a Q-algebra. Some properties of a cubic Q-ideal with a cubic BCK-ideal are studied, and a few results of these ideals are discussed. Such that every cubic Q-ideal is a cubic BCK-ideal, and the opposite is not true. Also, the concept of the level set of a Q-algebra as a cubic set is defined, and some important results associated with it have been shown. The relationship between a cubic Q-ideal and a Q-ideal with a cubic Q-sub-algebra and a Q-sub-algebra by the level set is proved. Finally, the image and the inverse image of cubic Q-ideals under a homomorphism are introduced, and a few theorems are proven, such that the image and the inverse image of a cubic Q-ideal are also cubic Q-ideals by the homomorphism mapping