In this paper, we introduce the concept of (prime) ideals on neutrosophic extended triplet groups (NETGs) and investigate some related properties of them. Firstly, we give characterizations of ideals generated by some subsets, which lead to a construction of a NETG by endowing the set consisting of all ideals with a special multiplication. In addition, we show that the set consisting of all ideals is a distributive lattice. Finally, by introducing the topological structure on the set of all prime ideals on NETGs, we obtain the necessary and sufficient conditions for the prime ideal space to become a $T_1$-space and a Hausdorff space.

We first introduce a new kind of algebras named left Ehoop, in which the top element “1” is changed as one of the maximal elements. Therefore, left Ehoops are generalizations of left hoops, which is one of the most fundamental fuzzy logic algebras as the semantical system of fuzzy logic system. Additionally, we explore the connections between left Ehoops and several related structures, including EL-algebras, CL-algebras, BCK-algebras, and quantum B-algebras. To further clarify the link between left Ehoops and EL-algebras, we also define the concept of self-similar EL-algebra. More importantly, we give an equivalent characterization such that left Ehoops become self-similar EL-algebras. Lastly, we define Bosbach states, Riečan states, and state-morphisms on left Ehoops, while examining their key properties and interrelationships. Our findings show that the collection of Bosbach states on a left Ehoop constitutes a compact Hausdorff space, and the mapping $\phi$ from state-measures to Bosbach states is an affine homeomorphism.

The selection of energy suppliers is important for sustainable energy management, as selecting the most appropriate suppliers reduces the environmental impact and improves resource optimization through sustainable practices. Our primary objective of this work was to develop a system for identifying energy suppliers by assessing various characteristics and their associated sub-attributes. Interval-valued q-rung orthopair fuzzy hypersoft sets (IVq-ROFHSS) originate by developing an association among interval-valued q-rung orthopair fuzzy sets and hypersoft sets. It is a crucial resource to handle unpredictable situations, mainly when presenting a component in a real-life scenario. IVq-ROFHSS is a new structure developed to manage the sub-parametric values of the alternatives. We developed the Einstein operational laws for IVq-ROFHSS and extended the Interval-valued q-rung ortho-pair fuzzy hypersoft Einstein weighted average (IVq-ROFHSEWA) and interval-valued q-rung ortho-pair fuzzy hypersoft Einstein weighted geometric (IVq-ROFHSEWG) operators. Moreover, we used the developed operators to formulate a multi-attribute group decision-making strategy to choose the ideal provider in sustainable energy management. The presented fuzzy robust approach reliably reiterated the challenged energy supplier selection in supply chain management to regular activities while alleviating overall expenses and promising stable reliability.