Serratus posterior inferior muscle
The serratus posterior inferior muscle is a 
muscle of the mid back that connects the rib cage to the spine. It attaches to the rib cage over the 9th to 12th ribs. It is divided up into four sections, and receives blood supply from the intercostal artery.
The serratus posterior inferior compresses the rib cage down and toward the spine. Having the rib cage pulled to the back allows a better range of motion for the upper body, giving it the ability to rotate and extend. This also assists in breathing, acting as a balance to the pull on the rib cage caused by the diaphragm.
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Origin
The serratus posterior inferior originates from the supraspinous ligament and the spinous processes of the last two thoracic vertebrae, T11 and T12, and the first two lumbar vertebrae, L1, and L2.
Insertion
The serratus posterior inferior inserts on the posterior side of the four lowest ribs, 9-12.
Word origin
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Serratus means saw tooth, in the same way as in serrated knife, referring to the back-and-forth pattern the muscle makes as it attaches to each rib.
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Posterior, meaning on the back side, distinguishes it from the serratus anterior muscle - a similar muscle on the front side of the rib cage.
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Inferior distinguishes it from the serratus posterior superior muscle, which sits above it.