Please note that the concept of a binary operation can be derived from the set theory. It is a function, which is a special type of a relation. On the other hand, a relation is a subset of the Cartesian product $X\times X.$ In the following, we define algebraic structures and also derive this concept directly from the set theory:
Let $X$ be a set, and let $\ast_1,\ast_2,\ldots,\ast_n$ be binary operations defined on it. Since latter are also sets, by definition, we can build an ordered $n$-tuple $(X,\ast_1,\ast_2,\ldots,\ast_n).$ Thus, this tuple is a set, called the algebraic structure (or algebra) of $X$ under the binary operations $\ast_1,\ast_2,\ldots,\ast_n.$
Chapters: 1 2
Corollaries: 3 4
Definitions: 5 6 7 8 9 10 11 12 13 14 15 16 17
Examples: 18
Explanations: 19
Parts: 20
Proofs: 21 22 23
Propositions: 24 25 26 27
