ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

In the previous part the subject was general algebraic structures. We made no
assumptions on the number or the character of the operations. Nor did we use
any laws of calculation. Now we turn to specific algebraic structures. These
are defined by specifying which types of operations are in play and which rules
(also known as axioms) they must obey. For each such specification there will
be many concrete algebraic structures answering the specification. We shall
speak about a category of algebraic structures when we consider the collection
of all structures determined by a given specification.
We shall advance gradually through different categories, in each step adding
some extra operation or some extra axiom.
Consequently we start out with the category of semigroups, where there is only
one operation and only one axiom, associativity.
In the next step we pick out an element (considered to be a 0-ary operation)
and claim as an axiom that this element is a neutral element. This is the
category of monoids.
We continue this way adding more operations and more axioms through the
categories of groups, rings and fields.
For each of these categories we study the induced structures and check if they
them selves belong to the category. Since they have the same type it is enough
to check if the axioms are satisfied.
Then we study homomorphisms between structures of the same category .
7: Semigroups
7. 1: Definition
59. Definition: Semigroup
An algebraic structure (A, ♢ ), where ♢ is binary and associative is said to be
a semigroup.
60. Remark: Semigroups are half groups
The name semigroup refers to the fact that many semigroups make up half (so
to speak) of some group, for instance the natural numbers with addition, which
is (roughly) half of the group of integers with addition. As a matter of fact it
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