Highlights• This paper focuses on the theories of dual-generalized and hyperbolic-generalized complex numbers.• The algebraic structures of dual-generalized and hyperbolic-generalized complex numbers are given.• Dual-generalized complex and hyperbolic-generalized complex valued functions are defined.• The matrix representations of dual-generalized and hyperbolic-generalized complex numbers are stated.• An efficient classification includes complex-generalized complex numbers is examined.

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This paper aims to develop dual-generalized complex Fibonacci and Lucas numbers and obtain recurrence relations. Fibonacci and Lucas's approach to dual-generalized complex numbers contains dual-complex, hyper-dual and dual-hyperbolic situations as special cases and allows general contributions to the literature for all real number p . For this purpose, Binet's formulas along with Tagiuri's, Hornsberger's, D'Ocagne's, Cassini's and Catalan's identities, are calculated for dual-generalized complex Fibonacci and Lucas numbers. Finally, the results are given, and the special cases for this unification are classified.

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In this paper, the definitions of soft Γ-semirings and soft sub Γ-semi rings are introduced with the aid of the concept of soft set theory introduced by Molodtsov. In the mean time, some of their properties and structural characteristics are investigated and discussed. Thereafter, several illustrative examples are given.