A lineal heir is someone who inherits in a line that ascends or descends from a common ancestor. The heir can be above or below the decedent in the direct line of descent. For example parents, children, grandchildren etc.
Example of a state statute on lineal heirs
[Official Code of Georgia Annotated; Title 44 Property, Chapter 6 Estates, Article 2 Fee Simple Estates]
O.C.G.A. § 44-6-23. Construction of words such as "heirs" or "heirs of body."
Limitations over to "heirs," "heirs of the body," "lineal heirs," "lawful heirs," "issue," or words of similar meaning shall be held to mean "children" whether the parents are alive or dead. Under such words the children and the descendants of deceased children by representation in being at the time of the vesting of the estate shall take.