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In this paper, we propose a cognitive radio-inspired rate-splitting multiple access (CR-RSMA) scheme to assist semi-grant-free (SGF) transmissions in which a grant-based user (GBU) and multiple grant-free users (GFUs) access the base-station (BS) by sharing the same resource block. Using the cognitive radio principle, the GBU and admitted GFU are treated as the primary and secondary users, respectively, and rate-splitting is applied at the admitted GFU to realize SGF transmissions. The admitted GFU's transmit power allocation, target rate allocation, and successive interference cancellation decoding order at the BS are jointly optimized to attain the maximum achievable rate for the admitted GFU without deteriorating the GBU's outage performance compared to orthogonal multiple access. Due to the extended non-outage zone, CR-RSMA-assised SGF (CR-RSMA-SGF) transmissions achieve a lower outage probability than SGF transmissions assisted by cognitive radio-inspired non-orthogonal multiple access. Exact expressions and asymptotic analysis for the admitted GFU's outage probability are derived to evaluate the system performance achieved by CR-RSMA-SGF transmissions. The superior outage performance and full multiuser diversity gain achieved by CR-RSMA-SGF transmissions are verified by the analytical and simulation results.