Algebraic Structures are important for the creation or study of a model for a subject matter under analysis. Moreover the Algebraic processes are one of the chief elements for the foundation of whole Mathematics; and the conception of Algebraic Structures surfaces just with the initial organization of Mathematics through the introduction of the ideas of Set, Mappings. In this post and other following posts, I will survey these Algebraic Structures; mostly guided by Nicolas Bourbaki’s books. The style of analysis of these Algebraic Structures will be done taking into cognizance the patterns present in them, the Formal Systems, and the expressiveness of these structures. The purpose of these posts is just to appreciate the methodology of organization of ideas that Mathematics provides through Algebra, and the author will pay little effort to give a formal introduction of elements of Algebra.

Nicolas Bourbaki have (the plural have is in context of the fact that “Nicolas Bourbaki”, in Algebraic realm, denotes group of people rather than a single person) defined Magma as such:-

Let be a set. A mapping of into is called a law of composition on . The value of for an ordered pair is called the composition of and under this law. A set with a law of composition is called a magma.

Thus if the underlying set for the model of a subject matter under study has a law of composition, then the abstraction which contains this set with the mentioned law of composition is eligible to be called a Magma. Since law of composition works on a single set, it is the simplest binary operation pattern on the set. We can think of another law of composition, , such that . The and in the ordered pair are just the placeholders, for a given mapping. How do and quantify is  subject to the illustration of . Here in the case of , the role of (and respective ) is same as that of in (and respective ). We have a  definition here:-

Let be a magma and denoted its law of composition. The law of composition on is called the opposite of the above. The set with this law is called the opposite magma of .

This shuffling of the placeholders to create an Opposite Magma, is the simplest of the tweaks on a law of composition. That for a given magma , we are looking into opposite magma , necessarily means that we are interested in a relation between and ; and of course the simplest of such relations will be . This hints of Commutativity, which we will discuss later on.

How we can link two Magmas ? That there exists some mapping between elements of sets of these Magmas tells about relation between the underlying sets, the wider question of relation between Magmas must talk about respective law of composition also. If we denote as the law of composition in each of these Magmas, say and ( usually will have different meaning for each Magma), and is the mapping from the first set to the second one, then if , we have the simplest relation between these two Magmas.
A mapping of into such that the relation holds for every ordered pair is called a homomorphism, or morphism, of into ; if , is called an endomorphism of .

The homomorphism will be called isomorphism, if is a bijective mapping. Homomorphism asserts stronger relation between two Magmas; the calculations for the laws of composition in second magma, can be effectively done in first magma, . Also any special property of law of composition in first magma will be reflected in second magma. Isomorphism asserts more. The two Magmas will have same behavior except for, possibly, the different names of elements and the law of composition. Of course some extra calculations may be needed( for example, of mapping function ), yet the structure of these two magmas are the same.
The exploration of Opposite Magma, Homomorphism, Isomorphism is some of the central activities in Algebra. In the next post, an extension of law of composition over a family of elements of a magma will be presented along with description of a formal system about it.