Enhanced Characterization of Rough Semigroup Ideals: Extension and Analysis

The principal and central concept of rough sets was taken up by Pawlak in the year 1982 [1]. It's a suitable tool for studying uncertain and imprecise knowledge in information systems in a mathematical way. The theory and applications of rough set theory have been extensively researched over the past few years. In rough set algebra, two additional approximation operators are added to the set algebra. It can therefore be said that rough set theory extends set theory. Rough sets play a fundamental role in artificial intelligence and cognitive sciences, especially in machine learning, data mining and knowledge discovery, and pattern recognition. Data that is imprecise and noisy can be analysed using rough set approaches.

Mathematically, a system consists of a set that establishes relations between its elements or subsets in some way. The system categorizes data based on partial order as a key categorizing factor. It uses any two elements of a set as well as any two subsets of a set. Each pair of elements in the set does not need to have it defined. This type of relation is referred to as a totally ordered.

It is possible to approximate any set with lower or upper approximation by dividing the set in the universe by using relations, either partial order relations or totally ordered relations. As we consider our universe as our system, it may be impossible to fit everyone into a single cell in our universe if we divide it up by characters which is why an approximate set of characteristics may exist for any group or individual, where the characters that are entirely appropriate for them are called lower approximation. The characters that are appropriate for them are called upper approximation. A rough set is one with non-empty differences between upper and lower approximations, otherwise it is crisp. So far, many results have been produced by various experts while evaluating the algebraic aspects of the rough set.

Bonikowsaki [2], Iwinski [3], Pomykala and Pomykala [4] and Biswas and Nanda [5] gave an algebraic approach to rough sets. Kuroki and Wang [6] evaluated rough group with respect to approximation of lower and upper bounds of any subset of group with respect to its normal subgroup.

Based on arbitrary binary relations, Liu and Zhu [7] presented the lower and upper approximations. Also, Pawlak and Skowron [8] provided some extensions to rough sets.

Using fuzzy ideals with thresholds in ordered semigroups, Hussain et al. [9] proposed generalized roughness for fuzzy filters. Hussain et al. [10] assessed the roughness of fuzzy ideals in ordered semigroups by using isotone and monotone mappings also gave the concept of rough Pythagorean fuzzy ideals in semigroups

In particular, in semigroups the idea of semigroup rough ideals was initiated by Kuroki [11]. Based on fuzzy ideals, Wang and Zhan [12] introduced the concept of rough semigroups. Bagırmaz and Ozcan [13] gave the concept of rough semigroups and its homomorphism, also introduced ideals and bi-ideals in rough set. Soft sets have been used to introduce roughness in semigroups by Arabi and Talebi [14]. An approach to generalized rough approximation spaces based on maximal neighbourhoods and ideals was given by Hosny et al. [15]. Guler et al. [16] provided rough approximations via ideals for various topologies. Sangeetha and Sathish [17] defined rough groups using upper and lower approximations to rough sets within a finite universe.

Our focus will be on roughness in semigroups ideal, semi lattices, and quotient semigroups, motivated by the study of roughness in algebraic systems and partially ordered sets. A rough semigroup ideal is extended in this paper along with results related to rough semilattices and rough quotient semigroups. With the help of appropriate examples, these concepts are explained.

In this paper, the organization is as follows: Section 2 covers rough sets and their basic concepts, in section 3, concepts of rough semigroups and rough ideals were discussed, in section 4, the concept of rough h-ideals was introduced, in section 5, rough semi-lattices and in section 6, rough quotient semigroups were discussed. Some results from these concepts have been proved and verified with relevant examples.

Definition 3.1 [5]

When a non-empty set is closed and associative under a binary operation, it is a semigroup and represented by (S, *).

Definition 3.2 Subsemigroup [5]

If AA $\subset$ A, for any subset A of S, then A is called sub semigroup.

Definition 3.3 [5]

Relation $\rho$ on the semigroup is a congruence relation if $a_1, b_1 \in S$ then $\left(a_1, b_1\right) \in \rho$ implies $\left(a_1 x, b_1 x\right) \in \rho \quad \&$ $\left(x a_1, x b_1\right) \in \rho$ for all $x \in S$. Furthermore, $\rho$ is said to be complete if $\left[a_1\right]_\rho\left[b_1\right]_\rho=\left[\begin{array}{ll}a_1 & b_1\end{array}\right]_\rho$.

Definition 3.4 [8]

A₁, a non-empty subset of S. Then:

$\cdot \rho_{A_1}=\left\{x_1 \in S \mid\left[x_1\right]_\rho \subseteq A_1\right\}$

$\cdot \rho^{A_1}=\left\{x_1 \in S \mid\left[x_1\right]_\rho \cap A_1 \neq \phi\right\}$

called $\rho$-approximations (lower $\&$ upper) of $A_l$ respectively. If $\rho^{A_1}-\rho_{A_1} \neq \phi$, then $A$ is rough with respect to $\rho$.

Definition 3.5 [8]

Let $\rho_1, \rho_2$ be complete congruence relations on $S$ and $A_1, B_1$ are non-empty subsets of $S$. The following results are due to Kuroki $[11\rceil$ :

$\begin{aligned} & \cdot \rho_{1_A} \rho_{1_B} \subseteq \rho_{1_{A B}} \\ & \cdot\left(\rho_1 \cap \rho_2\right)^A \subseteq \rho_1^A \cap \rho_2^A \\ & \cdot\left(\rho_1 \cup \rho_2\right)^A=\rho_1^A \cup \rho_2^A \\ & \cdot\left(\rho_1 \cap \rho_2\right)_A=\rho_1^A \cap \rho_2^A \\ & \cdot\left(\rho_1 \cup \rho_2\right)_A \supseteq \rho_1^A \cup \rho_2^A \\ & \cdot \rho_1 \subseteq \rho_2 \Rightarrow \rho_{1_A} \supseteq \rho_{2_B} \\ & \cdot \rho_1 \subseteq \rho_2 \Rightarrow \rho_1^A \subseteq \rho_2^B\end{aligned}$

Definition 3.6 [5]

If $B S \subseteq B$, then $B$ is an ideal (left), if $S B \subseteq B$, then $B$ is an ideal (right) of the semigroup and if both are true then $B$ is two-sided ideal.

Definition 3.7 Rough subsemigroup [8]

Let $S$ be a semigroup. A non-empty subset $A$ of $S$ is a rough upper (lower) sub semigroup if the corresponding $\rho^A\left(\rho_A\right)$ is sub semigroup of $S$ respectively. If both $\rho^A$ and $\rho_A$ is subsemigroup of $S$ then $A$ is rough sub semigroup of $S$.

Definition 3.8 Rough ideals [8]

If $\rho^{A_1}$ is a right [left, two-sided] ideal of $S$, then $A_l$ is rough upper right [left, two-sided] ideal of $S$. If $\rho_{A_1}$ is a right [left, two-sided] ideals of $S$, then $A_1$ is a rough lower right [left, two sided] ideal of $S$.

Definition 3.9

If A₁ is a subsemigroup of S then it is rough upper subsemigroup of S. Also, if A₁ is right [left, two-sided ideal] of S, then A₁ is rough upper right [left, two-sided ideal] of S.

Definition 5.1 Semilattices

Semilattices are defined as posets P satisfying the following characteristics.

·A least upper bound (lub) is given to each subset of two or more elements.

·A greatest lower bound (glb) is given to each subset of two or more elements.

Lower semilattices are those that satisfy only the second condition, while upper semilattices are those that satisfy only the first condition.

The powerset of any set is a semilattice with respect to partial order inclusion.  Here, we consider a collection of rough upper (lower) ideals (left/right) and it forms semilattice and hence called rough upper (lower) semi – lattice.

Definition 5.2 Rough semi – lattices

$\left\{A_1, A_2, \ldots, A_n\right\}$ are collection of finite set of ideals (left) of $S$, now the collection of rough upper ideals (left) of $S,\left\{\rho^{A_i}\right\}$ where $i=1,2, \ldots ., n$ forms rough upper semi-lattice with respect to partial order inclusion. In a similar manner rough lower semi-lattice defined.

With suitable examples, the following theorems 5.1 & 5.2 explain when rough upper ideals become rough semilattices.

Theorem 5.1

The set of rough upper ideal (left) of S forms a rough upper semilattice with respect to inclusion.

Proof:

Consider the finite collection of sets of rough upper ideal (left) of S.

If $A_1, A_2, \ldots, A_n$ are ideals (left) of $S \& \rho$, an equivalence relation on $S$.

Then $\rho^{A_1}, \rho^{A_2}, \ldots \rho^{A_n}$ are rough upper ideal (left) of $S$.

Since the union of rough upper ideal (left) of $S$ is a rough upper ideal (left) and smallest set containing all of them is its union.

Hence union of $\rho^{A_i} \quad i=1,2, . . n$ serves as least upper bound.

Hence it forms rough upper semilattice with respect to inclusion.

Theorem 5.2

The set of rough lower ideal (left) of S forms a rough lower semilattice with respect to inclusion.

Proof:

If A₁, A₂, …, A_n are ideals (left) of S & ρ, an equivalence relation on S.

Then $\rho_{A_1}, \rho_{A_2} \ldots . \rho_{A_n}$ are rough lower ideal (left) of $S$. (if it is non-empty).

Since $\phi$ is contained in any set. Infimum exists and $\phi$ serves as greatest lower bound and it forms a rough lower semilattice with respect to inclusion.

Example 5.1

Let S={1, 2, 3, 4} be a semigroup with respect to * given by

The ideals (left) of S are (1), (2), (4), (12), (14), (24), (123), (124), S and are represented by A₁, A₂, A₃…. A₉ respectively. These forms a semilattice with respect to inclusion.

Let $\rho$ congruence classes are given by $\{\{1,2,4\},\{3\}\}$.

$\rho^{A_i}=A_8$, for $i=1,2,3,4,5,6,8$

$\rho^{A_7} \& \rho^{A_9}=A_9$

$A_8$ and $A_9$ are rough upper ideal (left).

So, there is an upper semilattice whose lub is A₉ and glb is A₈. (Figure 1).

1.png

Figure 1. Hasse diagram

Similarly, lower approximation of above is given by

$\rho_{A_i}=\phi$, for $i=1,2,3,4,5,6,8$

$\rho_{A_7}=\{3\} \& \rho_{A_9}=A_9$

So, there is a lower semilattice whose lub is $A_9$ and glb is $\phi$. (Figure 2).

2.png

Figure 2. Hasse diagram

Example 5.2

S={1, 2, 3, 4, 5, 6}, along with binary operation * given by

The ideals (left) of S are (1), (12), (13), (15), (124), (123), (126) and S represented by A₁, A₂, A₃, A₄, A₅, A₆, A₇, A₈ respectively and it forms a semilattice with respect to inclusion.

Let $\rho$ congruence classes are $\{\{1,2,3\},\{3\},\{5,6\}\}$

$\rho^{A_i}=A_5$ for $i=1,2,5, \rho^{A_i}=A_8$ for $i=4,7,8$

$\rho^{A_3} \& \rho^{A_6}=\{1,2,3,4\}$

Hence, there is a rough upper semilattice whose lub is A₈ and glb is A₅ (Figure 3).

3.png

Figure 3. Hasse diagram

Similarly lower approximation is given by

$\rho_{A_i}=\phi$, for $i=1,2,4,7, \rho_{A_i}=\{3\}$ for $i=3,6, \rho_{A_5}=$ $\{12,4\}, \rho_{A_8}=A_8$ (Figure 4$)$.

4.png

Figure 4. Hasse diagram

Definition 5.3. Minimal rough upper ideal

Let $A$ be an ideal (right, left) of a semigroup. $\rho$, a relation(congruence) on $S$ and which divides $S$ into classes of equivalence. Now, $\rho^A$ called minimal rough upper ideal (right, left) of $S$ if there exists a $B \subseteq \rho^A$ then $\rho^B$ is not a rough upper (left, right) ideal otherwise $\rho^B=\rho^A$.

Theorem 5.3

Any minimal rough upper ideal (left) is a rough upper h-ideal (left) of a semigroup.

Proof:

If $\rho^A$, a minimal rough upper ideal (left), then $\rho^A S \subseteq \rho^A$.

If $B \subseteq \rho^A$ then $\rho^B$ never be a rough upper ideal (left) of $S$ otherwise $\rho^B=\rho^A$.

Consider $\rho^A S S=\left(\rho^A S\right) S \subseteq \rho^A S$

$\Rightarrow \rho^A S$ is rough upper ideal (left) of $S$.

Since $\rho^A$ is minimal rough upper ideal (left), $\rho^A \mathrm{~S}=\rho^A$

$\Rightarrow \rho^A$ is rough upper $h$-ideal (left) of $S$.

Theorem 5.4

If $\rho^A$ is any rough upper ideal (right) of $S \& \rho^B$ is minimal rough upper ideal (left) of $\rho^A$, then $\rho^B$ is rough upper $h$-ideal (left) of $S$.

Proof:

Since $\rho^A$ is any rough upper ideal (right) of $S$ then $S \rho^A \subseteq$ $\rho^A$.

Also, $\rho^B$ is minimal rough upper ideal (left) of $\rho^A$, then $\rho^B \rho^A \subseteq \rho^B$.

Now, $\rho^B S \rho^A=\rho^B\left(S \rho^A\right) \subseteq \rho^B \rho^A \subseteq \rho^B$.

Since $\rho^B$ is minimal rough upper ideal (left) of $\rho^A$. Hence $\rho^B \mathbf{S}=\rho^B$.

Theorem 5.5

Every minimal rough upper ideal (right) is a rough upper h-ideal (right).

Proof:

If $\rho^A$ is minimal rough upper ideal (right), then $S \rho^A \subseteq \rho^A$.

If $\boldsymbol{B} \subseteq \boldsymbol{\rho}^{\boldsymbol{A}}$, then $\rho^B$ is never be a rough upper ideal (right) otherwise $\rho^B=\rho^A$.

Now, $\boldsymbol{S} \boldsymbol{S} \boldsymbol{\rho}^A=\boldsymbol{S}\left(\boldsymbol{S} \boldsymbol{\rho}^A\right) \subseteq \boldsymbol{S} \boldsymbol{\rho}^A$

$\Rightarrow \mathrm{S} \rho^A$ rough upper ideal (right) of $S$ because of minimality of $\rho^A$.

$\mathrm{S} \rho^A=\rho^A$. Hence $\rho_m^A$ is rough upper $h$-ideal (right) of $S$.

The theory of rough sets can be applied to the algebraic system - semigroups. Rough upper h-ideals (left & right) in semigroups are an extension of h-ideals in semigroups, with some properties proved for them. We have also extended the results proposed by Kuroki [11], proved some new properties of rough upper ideal (left & right) of a semigroup. In rough quotient semigroups, Kuroki [11] proved that $\rho_1^A \rho_2^A \subseteq\left(\rho_1 . \rho_2\right)^A$ if A is a subsemigroup. We have extended the same for lower approximation of congruence classes if A is h- ideal (left) of S. The minimal rough upper ideal (left, right, h-ideal) has been discussed. As far as rough set algebraic structure is concerned, rough upper ideals (left, right, bi-ideal) have been employed so far, but here we have introduced rough h-ideals by combining rough set concepts with semigroup ideal concepts. A distinction is made between the relatively new ideas of rough set theory and those that are already well-established. The rough set theory and semigroup theory were connected via novel ideas.