LetKbe the restricted contact Lie algebras K(n, 1) over an algebraically closed field F of characteristic p >3. We prove that each skew-symmetric biderivation of K is inner and show that commuting maps on K are scalar multiplication maps. Moreover, it is showed that the commuting automorphisms and dertivations of K are proved to be the identity mappings and zero mappings, respectively. Meanwhile, the commutative post-Lie algebra structures on K are given.

Based on the definitions of fuzzy associative algebras and fuzzy ideals, it is proven that the intersections of fuzzy subalgebras are fuzzy subalgebras, and the intersections of fuzzy ideals are fuzzy ideals. Moreover, we prove that the $\mathrm{kernels}$ of fuzzy homomorphisms are fuzzy ideals. Using fuzzy ideals, the quotient structures of fuzzy associative algebras are constructed, their corresponding properties are discussed, and their homomorphism theorems are proven.

Pullbacks are fundamental limit constructions in category theory. Their study within intuitionistic fuzzy module categories, which provide a framework for handling mathematical uncertainty, remains underdeveloped. This work establishes their theoretical foundation in such categories. We first investigate the explicit construction of pullbacks and then study the properties of double pullbacks, monomorphisms, and epimorphisms. A classification of different pullback categories is provided. Key results include characterising pullback construction conditions, describing features of double pullbacks and related morphisms, and rigorously establishing the relationship between pullbacks and exactness. This work deepens the understanding of limit theory in intuitionistic fuzzy settings and provides a foundation for future research.