With the introduced properties of binary operations we are now able to define the first simple algebraic structures (or algebras). The following table shows which algebras will be introduced soon and which properties they require.
|Algebra $(X,\ast)$||Closure||Associativity||Neutral Element||Existence of Inverse||Cancellation||Commutativity.|
When in the table the entry "(✔)" is used, then it means that the defined binary operation "$\ast$" might not fulfill the property at all, which is required, if the sigh "✔" is used.
If an optional operation is fulfilled anyway, then the name of the algebraic structure is modified. For instance, if in a given semigroup the all elements $x,y\in X$ commute, i.e. $x\ast y=y\ast x,$ then we call such a semigroup a commutative semigroup. For some combinations, the name might even change. For instance, if $X$ is a monoid, and inverses exist for all elements of $X,$ then we have a group in place, but we will introduce groups in a later chapter. In this chapter, we want to concentrate on the tree simple-structure algebras: magma, semigroup, and monoid.