Unveiling New Dimensions in Soft Disemigraph Theory: Lexicographic and Restricted Lexicographic Products

Soft set theory provides a systematic approach for handling imprecision and uncertainty by categorizing elements of a set based on specific parameters. In semigraph theory, soft semigraphs utilize this approach, offering a parameterized perspective that has significantly advanced the field through effective parameter management. Building on this foundation, disemigraphs extend semigraphs by incorporating directional relationships among vertices, making them ideal for modeling scenarios where the sequence and direction of connections are crucial. In this paper, we introduce and explore the lexicographic product, and restricted lexicographic product of soft disemigraphs. We provide formal definitions, illustrative examples, and detailed proofs of several theorems to investigate the properties of these product operations.