In this paper, notions of ∈t-set and Qt-set of a fuzzy set f in a GE-algebra X are introduced and defined fuzzy GEalgebra in terms of ∈t-set. We provided conditions for the ∈t-set and Qt-set of a fuzzy set f to be GE-subalgebras of X. We provided conditions for a fuzzy set in a GE-algebra to be a fuzzy GE-algebra. The concept of fuzzy GE-filter of a GE-algebra is introduced and investigated its properties. We explored the conditions under which the ∈t-set and Qt-set can be GE-filters. Some characterizations of fuzzy GE-filters of GE-algebras are given.

In this paper, the properties of GE-ideals of transitive bordered GE-algebra are studied and characterizations of GE-ideals are given. We have observed that the set of all GEideals of a transitive bordered GE-algebra forms a complete lattice. The notion of bordered GE-morphism is introduced and established fundamental bordered GE-morphism theorem. A congruence relation on a bordered GE-algebra with respect to GE-ideal is introduced and some bordered GE-morphism theorems are derived.

The notions of ξ,-inside GE-derivation and ξ,-outside GE-derivation on a GEalgebra are introduced and its properties are investigated. Conditions for a self-map on GE-algebra to be a ξ,-inside GE-derivation and a ξ,-outside GE-derivation are provided. The ξ,-inside GE-derivation or the ξ,-outside GE-derivation ϱ,are used to form two sets X(ϱ, =ξ, ) and ker(ϱ, ), and GE-subalgebra and GE-, lter are studied for these two sets.

The concept of very true GE-algebra using very true operator is introduced and its properties are studied to expand the scope of research of GE-algebras. The concepts of simple very true GE-algebra and very true GE-filter are introduced. The characterization of simple very true GE-algebra is discussed, and several properties on very true GE-filter are investigated. Using a very true GE-filter, the quotient very true GE-algebra is constructed, and the uniform and topological space are established.

A new sub-structure called (vivid) deductive system is introduced and their properties are examined. Conditions for a subset to be a deductive system are provided. The notion of upper GE-set is also introduced, and an example to show that any upper GE-set may not be a deductive system are supplied. Conditions for an upper GE-set to be a deductive system are provided. An upper GE-set is used to consider conditions for a subset to be a deductive system. The characterization of deductive system is established, and relationship between deductive system and vivid deductive system are created. Conditions for a deductive system to be a vivid deductive system are given, and the extension property for vivid deductive system is constructed.

This paper presents a failure analysis of journal bearings used in GE-F9 gas turbines. Detailed studies including visual examination, optical microscopy, scanning electron microscopy, XRD and oil analysis were performed to determine the root causes of failure. Based on the results, it was determined that fretting, sulfur attack and fatigue were the main causes of failure.