Algebraic structures with
- an addition with the usual properties and
- an associative, but not necessarily commutative multiplication
- and with the distributive laws a(b+c) = ab+ac, (b+c)d = bd+cd
were studied in the late 19th and the 20th century
very carefully.
Since the construction (starting with a basis and defining on it a multiplication) imitated the creation of the complex numbers, such structures were said to be
Hypercomplex Systems.
Hypercomplex systems were often defined by generators and relations,
in this way one specifies the rules for calculating with specific variables.
The terminology was changed around 1930, when hypercomplex systems
were named
Algebras.