In fuzzy mathematical morphology, morphological operators are typically constructed using fixed structuring elements and fuzzy conjunctions. However, due to the diversity of image features, such fixed structuring elements struggle to adapt to local variations, thereby limiting the flexibility of algorithms. Moreover, the semi-overlap function, as a special type of aggregation functions that encompasses both continuous t-norms and overlap functions, has demonstrated broad theoretical significance and applications across multiple domains. In this paper, adaptive fuzzy mathematical morphology using semi-overlap functions is developed and applied to the field of image denoising: Firstly, fuzzy dilation and erosion operators are constructed by utilizing semi-overlap functions and their induced residual implications, and their algebraic properties are analyzed; Secondly, adaptive structuring elements are constructed by using a Gaussian weighted distance function and integrated with these morphological operators; Finally, Open--Close--Open method is developed to remove Gaussian noise from digital images. Compared with 6 existing methods, including fixed fuzzy mathematical morphology and adaptive mathematical morphology, the proposed method improves the comprehensive PSNR by 0.58 dB over the existing results.