The concept of an algebra being hom-associative is examined, and found to allow some awkward complications. A modified concept of strong hom-associativity is introduced to eliminate those quirks. It is proved that the basic “Yau twist” construction of a hom-associative algebra from an associative algebra does in fact produce strongly hom-associative algebras. It is proved that the axioms for a strongly hom-associative algebra yields a confluent rewrite system, and a basis for the free strongly hom-associative algebra is given a finite presentation through a parsing expression grammar.