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This paper defines the concept of partitioned hypergraphs‎, ‎and enumerates the number of these hypergraphs and discrete complete hypergraphs‎. ‎A positive equivalence relation is defined on hypergraphs‎, ‎this relation establishes a connection between hypergraphs and graphs‎. ‎Moreover‎, ‎we define the concept of (extended) derivable graph‎. ‎Then a connection between hypergraphs and (extended) derivable graphs was investigated‎. ‎Via the positive equivalence relation on hypergraphs‎, ‎we show that some special trees are derivable graph and complete graphs are self derivable graphs‎.

In this paper, we propose a new intuitionistic entropy measurement for multi-criteria decision-making (MCDM) problems. The entropy of an intuitionistic fuzzy set (IFS) measures uncertainty related to the data modelling as IFS. The entropy of fuzzy sets is widely used in decision support methods, where dealing with uncertain data grows in importance. The Complex Proportional Assessment (COPRAS) method identifies the preferences and ranking of decisional variants. It also allows for a more comprehensive analysis of complex decision-making problems, where many opposite criteria are observed. This approach allows us to minimize cost and maximize profit in the finally chosen decision (alternative). This paper presents a new entropy measurement for fuzzy intuitionistic sets and an application example using the IFS COPRAS method. The new entropy method was used in the decision-making process to calculate the objective weights. In addition, other entropy methods determining objective weights were also compared with the proposed approach. The presented results allow us to conclude that the new entropy measure can be applied to decision problems in uncertain data environments since the proposed entropy measure is stable and unambiguous.

The notions of a dual pseudo-Q algebra and a dual pseudo-QC algebra are introduced. The properties and characterizations of them are investigated. Conditions for a dual pseudo-Q algebra to be a dual pseudo-QC algebra are given. Commutative dual pseudo-QC algebras are considered. The interrelationships between dual pseudo-Q/QC algebras and other pseudo algebras are visualized in a diagram.

In this paper, we introduce the notions of multiplier, ⊙-closure operator and modal operator on hoop-algebras. After that, we investigate some algebraic properties of multiplier, ⊙-closure operator and modal operator on hoop-algebras. Then we study relationships between multiplier and closure operator on hoop-algebras showing that, any monotone modal operator on hoop-algebra serves as a closure operator. Finally we represent image of hoop-algebra under the closure operator can be a hoop-algebra.