تعریف عضو وارون

تعریف عضو وارون چپ و راست روشن است.

Definition. Let be a semigroup with identity e. Let a E S. If there exists an element b in S such that ab = e = ba,

then b is called the inverse of a, and a is said to be invertible.

Definition. A nonempty set G with a binary operation . on G is called a group jfthefollowing axioms hold:

(i) a(bc) = (ab)cfor all a,b,c E G. (ii) There exists e E G such that ea = afor all a E G. (iii) For every a E G there exists a' G such that a'a = e.

If the binary operation in a group (G,•) is commutative — that is. ab = ba for all a,b E G — then the group is called commutative or abelian.

تا اینجا: یک مجموعه که ضرب شرکت پذیر، عضو خنثی و وارون گروه. اگر ضرب جابجایی گروه آبلی

Let (S,') be a semigroup. Since- is associative, Theorem 4.2 (Chapter 1) holds here, and therefore there is a unique product of any given elements E S, in that order, that are written For any positive integer n we define, as before, a = aa...a (n-fold product), which is called the nih power of a. 

تعریف عضو خنثی

Definition. Let (S. ) be a semigroup. Ifthere is an element e inS such that ex—x=xe for aJixES,

then e is called the identity of the semigroup (S,).

تعریف right identity

 right identity مشابه left identity تعریف می‏ شود.

تعریف left identity

Let (S, ) be a semigroup and let a,b E S. We usually write ab instead of a b. An element e in S is called a left identity if ea=a for all a E S.

مثال هایی از نیم گروه

Examples of semigroups are

(a) The systems of integers, reals, or complex numbers under usual multiplication (or addition) (b) The set of mappings from a nonempty set S into itself under composition of mappings (c) The set of n X n matrices over complex numbers under multiplication (or addition) of matrices

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نحوه نمایش یک ساختمان جبری

Any algebraic structure S with a binary operation + or is normally written (S,+) or (S,•). However, it is also customary to use an expression such as "the algebraic structure S under addition or multiplication." Examples of semigroups are

ساده ترین ساختمان جبری

The simplest algebraic structure to recognize is a semigroup, which is defined as a nonempty setS with an associative binary operation. 

ساختارهای جبری مهم

Algebraic structures whose binary operations satisfy particularly important properties are semigroups, groups, rings, fields, modules, and so on. 

تعریف سیستم جبری

An algebraic structure or algebraic system is a nonempty set together with one or more binary operations on that set.