I am expressing my personal view without a historical excursus and being familiar with basic algebraic structures. Magma indicates a melted, liquid, formless substance. Algebraic magma requires closure property: result of an operation applied to a set is in the set. As for identity, inverse, associativity, commutativity, then magma does not require them. One can gradually and in different combinations add identity and/or other properties. This embeds "more structure": semigroup, group, ... Now, the algebraic magma is crystalizing yielding a structure. The origin is a formless bouillon from which the crystal of life is growing. I am not sure who has applied the term but it is a very accurate and impressive comparative image. A poet could envy.

I am expressing my personal view without a historical excursus. Geological magma associates with a melted, liquid, formless substance. 'Algebraic magma' requires only the closure property: result of an operation applied to a set is in the set. Citation: Bourbaki, N. Elements of Mathematics, Algebra I, Chapters 1-3. Paris: Hermann, Massachusetts: Addison-Wesley, 1974. Paragraph LAWS OF COMPOSITION, p. 1, Definition 1. "Let E be a set. A mapping f of E x E into E is called a law of composition on E. The value f(x, y) of f for an ordered pair (x, y ) [belonging to] E x E is called the composition of x and y under this law. A set with a law of composition is called a magma". As for identity, inverse, associativity, commutativity, then magma does not require them. Pretty unstructured! One begins gradually and in different combinations adding identity and/or other properties. This embeds "some structure": semigroup, monoid, group, ... The 'algebraic magma' is crystalizing! The origin is a "bouillon" from which the crystal of life is growing. I am not sure who has applied the term first time but it is a very impressive comparative image. A poet could envy.