**1. Introduction**

It is defined that a group is a set equipped with an operation described on it such that it has some properties as associated elements, an identity element, and inverse elements. Another definition can also be given as algebraic that the group is the set of all the permutations for algebraic expression's roots that displays the typical that the assembly of the permutations pertains to the set.

If questions are "how was group theory developed?, What is the importance of group theory in science or real life? investigated for the mathematical topic group theory, then we can understand easily why we work on the structures of the theory of the many types of groups.

As we know from the literature, some primary sources are determined in the development of group theory such as Algebra (Lagrange in the 17. century), Number Theory (Gauss in the 18. century) (Euler's product formula, Combinatorics, Fermat's Last Theorem, Class group, Regular primes, Burnside's lemma), Geometry (Klein, 1874), and Analysis (Lie, Poincaré, Klein in the 18. Century). It seems that three main areas have been described as Number Theory and Algebra (Galois theory, equation with degree 5, Class field theory), Geometry (Torus, Elliptic curves, Toric varieties, Resolution of singularities), and analysis in mathematics. Topology (((co)homology groups, homotopy groups) and algebraic part of it (Eilenberg–MacLane spaces, Torsion subgroups, Topological spaces), the Theory of Manifolds (manifolds with a metric), Algebra, Dynamical systems, Engineering (to create digital holograms), Combinatorial Number Theory, Mathematical Logic, Geometry in Riemannian Space, and Lie Algebra also belongs these three subjects.

Group theory is used not just in mathematics but also in computer science, physics, chemistry, engineering, and other sciences. Especially symmetry has a big potential property in the group theory. That is why it is considered as representation theory in physics. For example; mathematical works on quantum mechanics were done by von Neumann, Molecular Orbital Theory. Also, the Standard Model of particle physics, the equations of motion, or the energy eigenfunctions use group theory for their orbitals, classify crystal structures, Raman and infrared spectroscopy, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy or getting periodic tables-gauge theory, the Lorentz group-the Poincaré's group in modern chemistry or physics. Also, group theory is defined as representation theory in physics. A lot of groups with prime caliber built-in cryptography for elliptic curve do a service for public-key cryptography and Diffie–Hellman key exchange takes advantage of cyclic groups (especially finite) too. Additionally, cryptographic protocols also consider infinite nonabelian groups.

We can state the applications of the group theory also in real life as follows:


Thus, tools of the group theory are useful for working on applications in many different sciences and also real life as mentioned above.

Basic and simple examples can be given for usual groups as follows:


We can ask readers "whether or not these examples are partial group"? *Sm* is demonstrated by symmetric groups such that it includes *m*! permutations where *m* objects are taken from a set *A*. As an illustration, we can give the symmetric group *S*3. Supposing that *A* ¼ f g *a*, *b*,*c* and *S*<sup>3</sup> contains following objects; identity element*:*

*Algebraic Approximations to Partial Group Structures DOI: http://dx.doi.org/10.5772/intechopen.102146*

$$\text{Identity element} = \begin{pmatrix} a & b & c \\ a & b & c \end{pmatrix} \text{ and others are;}$$

$$\begin{pmatrix} \begin{pmatrix} a & b & c \\ b & a & c \end{pmatrix}, \begin{pmatrix} a & b & c \\ a & c & b \end{pmatrix}, \begin{pmatrix} a & b & c \\ c & b & a \end{pmatrix}, \begin{pmatrix} a & b & c \\ b & c & a \end{pmatrix}, \begin{pmatrix} a & b & a \\ c & a & b \end{pmatrix}$$

There are some properties in finite or infinite usual group theory. Some of them can be seen as follows:


If literature (briefly, references [1–47]) is investigated, then it is easily seen that partial groups are considered as topological structures more than algebraic structures. It is tried to prepare some new algebraic perspectives/approximations for the partial group. As we know, there are many algebraic infrastructures such as finiteinfinite group, abelian-nonabelian group, quaternion group, symmetric group, cyclic group, simple group, free group, orbits and stabilizers of the group, Lie group, and various kinds of theorems such as Sylow theorems, Cauchy theorem, Lagrange theorem, Cayley theorem, Isomorphism theorems as well as actions of groups for usual group theory.

An effect algebra is introduced in the foundations of mechanics [1]. Furthermore, effect algebra subjects are fundamental in fuzzy probability theory [2, 3]. Also, partial group is defined by [4] and used for topological and homological investigations. A pregroup can be defined as following:

A pregroup, [5]**,** of a set *P* containing an element 1, each element *p*∈ *P* has a unique inverse *p*�<sup>1</sup> and to each pair of elements *p*, *t*∈ *P* there is defined at most one product *pt* ∈ *P* so that;

a. 1 ∗ *p* = *p* ∗ 1 ¼ *p* is always defined,

b. *p* \* *p*�<sup>1</sup> = *p*�<sup>1</sup> \* *p* = 1 is always defined,

c. If *p* ∗ *t* is defined then *t* �<sup>1</sup> ∗ *p* �<sup>1</sup> is defined and equal to (*p* ∗ *t*) �1 .


Every pregroup is a partial group, but the converse is not true in general. In that meaning a partial group definition can be stated as follows:

A set *P* is a partial group in the meaning of ([6], Lemma 4.2.5) if each associated pair ð Þ *x*, *y* ∈ *PXP* there is at most one product *x:y* so that:

1.There is an element 1 ∈ *P* satisfying x.1 = 1. *x* ¼ *x* for each *x*∈*P*.

2.For each *x*∈*P* there exists an element *x*�<sup>1</sup> so that *x*.*x*�<sup>1</sup> =*x*�<sup>1</sup> .*x*=1

3. If *x:y* ¼ *z* is defined so is *y* �1 .*x*�<sup>1</sup> = z�<sup>1</sup> .

Inspiring by the groupoid and effect algebra [7] gave an alternative partial group definition as an algebraic style. They introduced partial subgroup, partial group homomorphism, etc. as an analog investigation for group theory. Moreover, readers can learn/consider a lot more structural results on the subject from others [8–47].

In this work, some (remained ones can be considered from readers using our works) fundamental results are given for partial groups. Similarities and differences have been noticed between usual groups and partial groups. Several of them also are described in this work.

Also, we do further investigations for partial groups in algebraic style. We define partial normal subgroup and give isomorphism theorems for partial groups. This work is important because it has both topological and algebraic applications. It can be expanded to rings or other algebraic structures.

### **2. Preliminary results**

Recently, an algebraic structure named as partial group (also known as Clifford Semigroup is isomorphic to an explicit partial group of partial mappings and it is a semigroup with central idempotents) is investigated with new structures in the literature.

A partial group (Clifford semigroup) is a regular semigroup (it means if M is a semigroup of group G then idempotent elements of M exchange with H's all elements). Another definition can be given for the partial group as "A regular semigroup with its central idempotents is named by Clifford semigroup."

Additionally, several partial algebras such as partial monoid, partial ring, partial group ring, partial quasigroup etc. … have been worked. For example, Jordan Holder Theorem of composition series is known to hold in every abelian category. The classical theory of subnormal series, refinements, and composition series in groups is extended to the class of partial groups which is known to be precisely the classes of Clifford semigroups, or equivalently semilattices of groups. Also, relations among the language theory, words, partial groups, universal group, and homology theory have been considered with an arrow diagram of the partial group.

In this chapter, we first state some basic properties of partial groups which are mentioned in several references.

**Definition 2.1** [7] Suppose *G*\* is a nonempty set: *G*\* is called as a partial group if the following conditions hold for all *x*, *y*, and *z*∈ *G* \* :

(G1) If *xy*,ð Þ *xy z*, *yz and x yz* ð Þ are defined, then the equality ð Þ *xy z* ¼ *x yz* ð Þ is valid.

(G2) For each *x*∈ *G*\* , there exists an e∈ G\* such that *xe* and *ex* are defined and the equality *xe* ¼ *ex* ¼ *x* is valid.

(G3) For each x∈ *G*\* , there exists an *x*'∈ *G*\* such that *xx*' and *x*'*x* are defined and the equality *xx*' ¼ *x*'*x* ¼ *e* is valid.

The *e*∈ *G*\* satisfies (G2) is called the identity element of *G*\* and the *X*'∈ *G*\* satisfies (G3) is called the inverse of *x* and denoted by *x*�<sup>1</sup> .

Another way, we can give partial definition as follows:

**Definition 2.2.** Let M be a semigroup. It is called *a* partial group if the followings are held.

i. Every *k*∈M has a partial identity *ek*

ii. Every *k*∈M has a partial inverse *k*�<sup>1</sup>


Note:


**Definition 2.3.** The regular element of the M semigroup is defined if there exists s ∈M such that ysy = y. Each element of M is regular element also M is named by regular semigroup.

**Definition 2.4.** M semigroup is called as completely regular semigroup for every element s ∈M ysy = y and ys = sy are satisfied.

**Note.** Unions of groups give us completely regular semigroups which are named by Clifford semigroups.

**Definition 2.5.** Let M be a semigroup. Elements k and s of a semigroup M are said to be inverse of each other if and only if sks = s and ksk = k.

Then following theorem can be given from the literature.

**Theorem 2.1.** The following results are equivalent to each other for a semigroupM:

i. M is a Clifford semigroup,

ii. There exists r ∈ S such that wrw = w and wr = rw for every w ∈M,

iii. M is a semilattice of groups,

iv. M is a completely regular inverse semigroup.

**Proposition 2.1.** M is a completely regular semigroup iff there are *ek*and *k*�<sup>1</sup> for every *k ϵ* M.

**Proposition 2.2** [7] Every group is a partial group and every partial group which is closed under its partial group operation is a group.

**Proposition 2.3.** Assuming that M is a partial group. Then, the following are given:


**Example 2.1** [7] Following sets with the given operation, can be seen as an example to the partial groups:


**Definition 2.6** [7] Suppose *G*\* be a partial group, *m* ∈*Z*+, and *a*∈ G\* . If *a*<sup>m</sup> is defined and *<sup>m</sup>* is the least integer such that *<sup>a</sup>*m<sup>¼</sup> *<sup>e</sup>*, the number *<sup>m</sup>* is called the order of *a*. In this case, it is called that *a* has a finite order element. If there does not exist an *<sup>m</sup>* <sup>∈</sup>*Z*<sup>+</sup> such that *<sup>a</sup>*m<sup>¼</sup> *<sup>e</sup>*, (if only *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*); then it is called that *<sup>a</sup>* has infinite order. The order of a is denoted by j j *a* .

**Example 2.2** [7] *<sup>G</sup>* <sup>¼</sup> 1, �1, *<sup>i</sup>*, �*i*, 2*i*, � *<sup>i</sup>* 2 with the multiplication operation on is a partial group and ∣*i*∣ ¼ 4, 2j j*i* ¼ ∞.

**Definition 2.7** [7]**.** Suppose *G*\* be a partial group. ð*<sup>G</sup>* \* Þ ¼ <sup>f</sup>*x*<sup>∈</sup> *<sup>G</sup>* \* <sup>∣</sup>*If ax and xa are defined for all a*<sup>∈</sup> *<sup>A</sup>*; *ax* <sup>¼</sup> *xa*<sup>g</sup> is called the center of *<sup>G</sup>*\* .

**Lemma 2.1** [7] A partial group is called *centerless* if Z (G\* ) is trivial i.e., consists of only the identity element. If G\* is commutative then G\* = Z (G\* ).

**Definition 2.8.** Supposing that M ¼ *Gp* be a partial group and ¼ *Hp* be a subset of M*:* is called by sub partial group of M if is a sub semigroup of M and *ek*, *k*�<sup>1</sup> are in for all *k*∈.

Especially, M and the set of idempotents elements of M are sub partial groups of M.

**Definition 2.9** [7] Let *G*\* be a partial group and *H*\* be a nonempty subset of *G*\* . If H\* is a partial group with the operation in *G*\* then *H*\* is called a partial subgroup of *G*\* .

**Example 2.3** [7] In Example 2.2 the set G\* <sup>¼</sup> f g 0, �1, … , �*<sup>n</sup>* where n*ϵ*<sup>+</sup> and + be known addition operation on is a partial group and let *H*\*¼ f g 0, �1, … , �*k* where 0≤*k*≤*n* and k∈. Then *H*\* is a partial subgroup of *G*\* .

**Lemma 2.2** [7] Let *G*\* be a partial group and H\* be a nonempty subset of *G*\* .*H* \* is a partial subgroup of *G*\* if and only if the following conditions hold:

i. *e*∈ *H*\* ;

ii. *a*�1∈ *H*\* for all *a*∈ *H*\* .

Moreover, Let *G* \* be a partial group and let *a* be an element of *G*\* such that the elements {ak for all k<sup>∈</sup> } are defined. Denote <sup>f</sup>*a*<sup>k</sup> ; *k*∈g ¼ <*a*> It is clear that the *Algebraic Approximations to Partial Group Structures DOI: http://dx.doi.org/10.5772/intechopen.102146*


**Table 1.**

*(G) is a partial group even it has not a group structure.*

set < *a*> is a partial subgroup of *G*\* . The partial subgroup <*a*> of *G*\* is called the cyclic partial subgroup generated by *a*. If there exists an element *a* in *G*\* such that <sup>&</sup>lt; *<sup>a</sup>*<sup>&</sup>gt; <sup>¼</sup> *<sup>G</sup>* \* , then *G*\* is called a cyclic partial group.

**Example 2.4** [7] Let *G* ¼ f g *e*, *a*, *b*,*c* ⊂ *S* ¼ f g *e*, *a*, *b*,*c*, *d* and "*:*" be a partially defined operation on G as in **Table 1**.

**Remark.** Note that *c:b* is undefined. Then *G* is not a group but it is a partial group. Additionally, in this partial group, <*a*> ¼ *G*, *G* is the cyclic partial group. But all partial subgroups of a cyclic partial group can not be cyclic. For instance, the partial subgroup *H* ¼ f g *e*, *a*,*c* is not cyclic. But in group theory, if a group is cyclic, all subgroups of it are also cyclic. The partial groups are different from groups in that meaning.

**Definition 2.10.** Assume that M ¼ *Gp* be *a* partial group and *k*∈M*:* Then, we define M*<sup>k</sup>* ¼ f g *s* ∈M : *ek* ¼ *es* .

**Theorem 2.2.** Suppose that M is *a* partial group and *k*∈M*:* Then, M*<sup>k</sup>* is a maximal subgroup M of which has identity *ek* and M ¼ ⋃f g M*<sup>k</sup>* : *k*∈M .

**Definition 1.11** [7] Let M and N be partial groups. A function *σ* : M⟶N is called a partial group homomorphism if for all *a*, *b*∈M such that *ab* is defined in M, *σ*ð Þ *a σ*(*b*) is defined in N and

$$
\sigma(ab) = \sigma(a)\sigma(b).
$$

If *σ* is injective as a map of sets, *σ* is said to be a monomorphism. If *σ* is surjective, *σ* is called an epimorphism.

**Definition 2.12.** For a partial group homomorphism *σ* : M⟶N it is defined ker*σ* ¼ *k*∈M : *σ*ð Þ¼ *k eσ*ð Þ*<sup>k</sup>* and *Im<sup>σ</sup>* <sup>¼</sup> f g *<sup>σ</sup>*ð Þ*<sup>k</sup>* : *<sup>k</sup>*∈<sup>M</sup> . Also, *<sup>σ</sup>* : <sup>M</sup>⟶<sup>N</sup> is named isomorphism if it is bijective.

As a consequence of the definition the following lemmas can be given:

**Definition 2.13.** Suppose that *σ* : M⟶N be a partial group homomorphism. Then, we define ker *σ* ¼ *k*∈M : *σ*ð Þ¼ *k e<sup>σ</sup>*ð Þ*<sup>k</sup>* and *Im<sup>σ</sup>* <sup>¼</sup> f g *<sup>σ</sup>*ð Þ*<sup>k</sup>* : *<sup>k</sup>*∈<sup>M</sup> .

**Theorem 2.3.** Assuming that *σ* : M⟶N be a partial group homomorphism and *k*∈M. Then, the following are given.

i. *σ*ð Þ¼ *ek e<sup>σ</sup>*ð Þ*<sup>k</sup> :*

$$\text{ii. } \sigma(k^{-1}) = \sigma(k)^{-1}.$$

iii. ker*σ is a subpartial group of* M*:*

iv. *Imσ is a subpartial group of* N.

v. *σ*ð Þ M*<sup>k</sup> is a subpartial group of* N*<sup>σ</sup>*ð Þ*<sup>k</sup> :*

vi. *σ*�<sup>1</sup> N*ek is a subpartial group of* M.

**Proposition 2.4** [7] Suppose M, N be partial groups and *σ* : M⟶N be a homomorphism of partial groups. Then the following conditions are satisfied:

i. If *A* is a partial subgroup of M, then *σ*(*A*) is a partial subgroup of N.

ii. If *B* is a partial subgroup of N, then *σ*�<sup>1</sup> (B) is a partial subgroup of M.

**Proposition 2.5.** Let *σ* : M⟶N be a homomorphism of partial groups. Then, it is obtained that

$$
\sigma(e\_k) = e\_{\sigma(k)}, \\
\sigma(k^{-1}) = \left(\sigma(k)\right)^{-1} \text{ for all } k \in \mathcal{M}.
$$

**Definition 2.14.** A sub partial group ¼ *Hp* of M ¼ *Gp* is named wide if the set of idempotent elements of Mis a subset of ¼ *Hp* and normal, written ⊲M. (if it is wide and *<sup>k</sup>k*�<sup>1</sup> <sup>⊆</sup> for all *<sup>k</sup> <sup>ϵ</sup>* <sup>M</sup>Þ*:*It is also trivial that the set of idempotents elements of M is a normal subgroup of M and it is called the set of idempotents elements of M the trivial normal subpartial group of M.

**Theorem 2.4.** Assuming that M be a partial group and *k*∈M, then

i. M*<sup>k</sup>* is a maximal subgroup of M with identity *ek*.

ii. <sup>M</sup> <sup>¼</sup> <sup>⋃</sup>f g <sup>M</sup>*<sup>k</sup>* : *<sup>k</sup>*∈<sup>M</sup> <sup>¼</sup> <sup>⋃</sup> <sup>M</sup>*ek* : *ek is* in the set of idempotents elements of Mg.

**Theorem 2.5.** If M is a partial group, then the set of idempotents elements of M is commutative and central.

**Definition 2.15.** For a partial group homomorphism*σ* : M⟶N it is defined ker*σ* ¼ *k*∈M : *σ*ð Þ¼ *k eσ*ð Þ*<sup>k</sup>* and *Im<sup>σ</sup>* <sup>¼</sup> f g *<sup>σ</sup>*ð Þ*<sup>k</sup>* : *<sup>k</sup>*∈<sup>M</sup> . Also, *<sup>σ</sup>* : <sup>M</sup>⟶<sup>N</sup> is named isomorphism if it is bijective.

**Definition 2.16.** *σ* is named as idempotent separating if *σ*ð Þ¼ *ek σ*ð Þ *es* implies that *ek* ¼ *es*, where *ek*,*es* are in the set of idempotent elements of M for a partial group homomorphism *σ* : M⟶N .

## **3. Partial normal subgroups**

In group theory, normal subgroup plays an important role in the classification of groups and gives lots of algebraic results. Now, we will construct an analog definition for partial groups. Throughout *G*<sup>p</sup> will denote the partial group. In this chapter, we should notice that if *G* is a group, then *G* is a partial group with *fect e*f g. Also, [8] in a group every element has a unique inverse, but in partial groups [7] for every element *a*∈ *G*, we have *Inv a*ð Þ 6¼ 0 because of that reason the identity element of the group differs from the identity element of the partial group. We can continue to work under these assumptions. From here on in, we will use the notation *G*<sup>p</sup> for partial groups.

**Definition 3.1 (Partial Conjugation Criteria)**. Let *G*<sup>p</sup> be a partial group, the element gp.*x*p.gp �<sup>1</sup> (or gp �1 .xp.gp) is called partial conjugate of *x*<sup>p</sup> by gp for fixed gp,*x*p∈ *G* p.

**Theorem 3.1.** Let *G*<sup>p</sup> be a partial group and Np be a partial subgroup of *G*<sup>p</sup> then following conditions are satisfied:

i. *N*<sup>p</sup> is normal in *G*<sup>p</sup> if and only if for all *x*p∈ *N*<sup>p</sup> and gp∈ *G*<sup>p</sup> we have gp �1 .*x*p. gp∈ *N*<sup>p</sup>

ii. *N*<sup>p</sup> is normal in *G*<sup>p</sup> if and only if for every element of *N*<sup>p</sup> all partial conjugates of that element also lie in *N*p.
