Sangeetha Sivakumar
| Shakeela Sathish*
© 2024 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
Rough set theory (RST) is a formal theory derived from logical properties of information systems. Rough set theory extends traditional set theory by defining a subset of a universe through the use of a pair of sets referred to as the lower and upper approximations. It is a mathematical approach for dealing with ambiguities and imprecisions in a variety of situation. Since its introduction by Zdislaw Pawlak in the late eighties of the previous century, it has evolved into pure and applied directions from mathematical, logical, and computational perspectives. The area of rough set theory in computational mathematics is rapidly developing. As far as vagueness and imprecision are concerned, rough set theory is basically a mathematical approach. An equivalence relation is a key concept in rough set models. Approximations at the lower and upper levels are constructed based on equivalence classes. There is wide application of algebraic systems in sequential machines, formal languages, arithmetic codes, and error-correction algorithms. The study of any set will be effective if an algebraic structure is developed for it. In the context of semigroups research, rough set theory can be used to analyse and understand the properties and relationships within semigroups. Semigroups and related algebraic structures and their properties can be explored more deeply when rough set theory is applied. The aim of this paper is to extend the concept of rough semigroup ideals. It has already been shown that some properties of rough (left, right) ideals in semigroups can be obtained by extending the notion of a left (right) ideal in a semigroup. As a result of considering h-ideals in semigroups, rough upper h-ideals (left & right) have been introduced here along with their properties. Also, the results related to rough semi-lattices and rough quotient semigroups are given. These concepts are explained with suitable examples.
rough semigroup, rough semigroups ideal, rough h-ideal, rough semilattices, rough quotient semigroup
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 2.1 [1]
An approximation space is composed of a set with finite elements Ω, called universe along with an equivalence relation (a relation which satisfies reflextive, symmetric and transitive properties) ∼ on Ω and it is represented by K=(Ω, ∼).
Definition 2.2 [1]
A family of subsets F={C1, C2, C3, ……, Cn} of Ω are said to be a classification of Ω if:
$\cdot C_1 \cup C_2 \cup ...... \cup \,C_n=\Omega$
$\cdot C_i \cap C_j=\phi$, for $\mathrm{i} \neq \mathrm{j}$
Also, in approximation space and for an element k in the universe, ∼ induces a class of equivalence [k]$\sim$.
Definition 2.3 [1]
Consider an approximation space where A is any subset of universe, then:
$\cdot \Omega^A=\left\{a_i \mid\left[a_i\right]_{\sim} \cap A \neq \phi\right\}$
$\cdot \Omega_A=\left\{a_i \mid\left[a_i\right]_{\sim} \subseteq A\right\}$
$\cdot B N_A=\Omega^A-\Omega_A$
are called approximations of upper, lower & boundary region of A with respect to ∼ respectively and A is said to be rough if BNA is non empty otherwise it is crisp.
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 4.1 Rough upper ideal
Let $A$ be any non-empty subset of a semigroup $S$ and $\rho$ be a congruence relation on $S$ also, $\rho^A$ is a subsemigroup of $S . A$ is said to be rough upper ideal (left) if $\rho^A S \subseteq \rho^A$ and rough upper ideal (right) if $S \rho^A \subseteq \rho^A$ and it is rough upper ideal if it satisfies the both conditions.
Definition 4.2 Rough upper h-ideal
$A$ is said to be rough upper h-ideal (left) if $\rho^A S=\rho^A$ and rough upper h-ideal (right) if $S \rho^A=\rho^A$ and it is rough upper h-ideal if it satisfies the both conditions.
Theorem 4.1
If X1 is a sub semigroup, h-ideal(left) of the semigroup, then X1 is rough upper h-ideal(left).
Proof:
Given $X_l, h$-ideal(left) of the semigroup, so $X_l \mathrm{~S}=X_l$. $X_1 S \subseteq X_1 \& X_1 \subseteq X_1 S$.
$\Rightarrow X_l$ is ideal (left) of $S$. Also, we have $\rho^{X_1} \rho^S \subseteq \rho^{X_1 S}$. Since, $\rho^S=S$ and by above, $\rho^{X_1} S \subseteq \rho^{X_1}$
$\Rightarrow X_1$ is rough upper ideal (left) of the semigroup $S$. Conversely, let $x \in \rho^{X_1}$, also $x \in \rho^S=S$.
Since $X_1 S \subseteq X_1, x \in \rho^{X_1} S$ completes the converse.
Theorem 4.2
If X1 is a sub semigroup, h-ideal(right) of the semigroup, then X1 is rough upper h-ideal(right).
Proof:
Given X1, h- ideal(right) of the semigroup, then SX1=X1
$S X_1 \subseteq X_1 \& X_1 \subseteq S X_1$
$\Rightarrow X_1$, ideal(right) of $S$. Also, we have $\rho^S \rho^{X_1} \subseteq \rho^{S X_1}$.
Since, $\rho^S=S$ and by above, $S \rho^{X_1} \subseteq \rho^{X_1}$.
$\Rightarrow X_1$ is rough upper ideal (right) of $S$.
Conversely, $x \in \rho^{X_1}$, also $x \in \rho^S=S$.
Since $S X_1 \subseteq X_1$ which implies $x \in S \rho^{X_1}$ proves the converse.
Example 4.1
$S=\{2,3,1\}$, semigroup with binary operation $*$ given by:
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(2), (23), (21) & S are h-ideals of S and are represented by hi1, hi2, hi3 & hi4 respectively.
$\rho$ - congruence classes are given by $\{\{2\},\{3\},\{1\}\}$ then $\rho^{h i_1}=h i_1, \rho^{h i_2}=h i_2, \rho^{h i_3}=h i_3, \rho^{h i_4}=h i_4$.
Hence $h i_1, h i_2, h i_3$ and $h i_4$ are rough upper h-ideal of $\mathrm{S}$.
Also, $\rho_{h i_1}=h i_1, \rho_{h i_2}, \rho_{h i_3} \& \rho_{h i_4}$ are empty, which implies $h i_1$ is rough lower $h$-ideal.
Example 4.2
$S=\{2,3,1\}$, a semigroup with binary operation $*$ given by:
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(2), (23), & S are h-ideals of S and are represented by hi1, hi2 & hi3 respectively.
$\rho$-congruence classes are $\{\{2\},\{3\},\{1\}\}$ then $\rho^{h i_1}=$ $h i_1, \rho^{h i_2}=h i_2, \rho^{h i_3}=h i_3$.
Hence $h i_1, h i_2 \& h i_3$ are rough upper $h$ ideal of $S$.
Also, $\rho_{h i_1}=h i_1$ and $\rho_{h i_2}, \rho_{h i_3}$ are empty, which implies $h i_l$ is rough lower $h$-ideal.
Example 4.3
$S=\{1,2,3\}$, a semigroup with binary operation $*$ given by:
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The h-ideals of S are (1), (13), & S represented as Z1, Z2 & Z3 respectively.
$\rho$ congruence classes are given by $\{\{1,2\},\{3\}\}$ then $\rho^{Z_1}=$ $\{1,2\}, \rho^{Z_2}=Z_3, \rho^{Z_3}=Z_3$.
Hence $Z_2 \& Z_3$ are rough upper $h$-ideal.
Now, $\rho_{Z_1} \& \rho_{Z_2}$ are empty, $\rho_{Z_3}=Z_3$, which implies $Z_3$ is rough lower $h$-ideal of semigroup.
Example 4.4
$S=\{1,2,3\}$ with binary operation $*$ given by:
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The h-ideals of S are (12) & S represented as X1 & X2 respectively.
$\rho$ congruence classes are given by $\{\{1\},\{2,3\}\}$ then $\rho^{X_1}=$ $X_1, \rho^{X_2}=X_2$.
Hence rough upper $h$-ideal of $S$ are $X_1 \& X_2$.
$\rho_{X_1}$ is empty & $\rho_{X_2}=X_2$, which implies $A_2$ is rough lower $h$-ideal of $S$.
Corollary 4.1
ρ1, a congruence relation on a semigroup and B1is h-ideal (right, left) of S. Hence B1 is rough upper h-ideal (right, left) but there is no need to hold the converse.
Example 4.5
$S=\{0,1,2,3\}$ along with binary operation $*$ given by:
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$\rho$ congruence classes are given by $\{\{0,3\},\{1\},\{2\}\}$ then $A_1=\{3\} \subseteq S, \rho^{A_1}=\{0,3\}, \rho^{A_1} S=\rho^{A_1}$
which implies $A_l$ is rough upper $h$-ideal (left) of the semigroup. But $A_l \mathrm{~S} \neq A_l$, it is not $h$-ideal (left) of $S$.
Theorem 4.3
If $A_1, B_1$ are $h$-ideals [left] of $S$, then $\rho^{A_1} \rho^{B_1}$ is rough upper $h$-ideal [left].
Proof:
We have A1S=A1, B1S= B1.
Also, A1 and B1 are rough upper ideal (left) of S.
(Theorem 4.1)
$\rho^{A_1} S=\rho^{A_1} \& \rho^{B_1} S=\rho^{B_1}$.
Now, $\rho^{A_1} \rho^{B_1} S=\rho^{A_1}\left(\rho^{B_1} S\right)=\rho^{A_1} \rho^{B_1}$.
(subsets of S holds the associative property)
Hence $\rho^{A_1} \rho^{B_1}$ is rough upper $h$-ideal [left] of $S$.
Theorem 4.4
If A, B are h-ideals [right] of the semigroup S then ρAρB is rough upper h-ideal [right] of S.
Proof:
We have A=SA & B=SB.
Also, both are rough upper ideal (right) of S. (Theorem 4.2)
$S \rho^A=\rho^A, \mathbf{S} \rho^B=\rho^B$.
Now, $S \rho^A \rho^B=\left(\mathrm{S} \rho^A\right) \rho^B=\rho^A \rho^B$.
$\Rightarrow \rho^A \rho^B$ is rough upper $h$-ideal [right] of the given semigroup.
Theorem 4.5
The collection of all rough upper ideals(left) of the semigroup forms rough upper ideal (right) of the set of all congruence subsets of the semigroup.
Proof:
A, an ideal(left) of the given semigroup.
$\Rightarrow \rho^A$ is rough upper ideal (left).
Denote $\boldsymbol{\Theta}$ as a collection of possible rough upper ideal (left) and $\eta$, a congruence class of all possible subsets of semigroup.
Now $\rho^A \in \Theta, B \in \eta$.
Now, $\left(B \rho^A\right) S=B\left(\rho^A S\right) \subseteq B \rho^A$, hence $\eta \, \Theta \subseteq \Theta$.
Theorem 4.6
The collection of all rough upper h-ideals (left) of the semigroup forms rough upper ideal (right) of the set of all congruence subsets of the semigroup S.
Proof:
Denote $\Pi$ as the collection of all rough upper $h$-ideal (left) of the semigroup and $\zeta$, congruence class of all subsets of the semigroup.
Let $A$ be any $h$-ideal which implies $\rho^A \in \Pi, B \in \zeta$.
Now, $\left(B \rho^A\right) S=B\left(\rho^A S\right)=B \rho^A$, hence $\zeta \Pi \subseteq \Pi$.
Theorem 4.7
For any rough upper ideal (left) X of semigroup, if Y is any h-ideal (left) of A, then ρY is rough upper ideal (left).
Proof:
Let $X$ be any ideal (left) of the semigroup, we have $X S \subseteq X$
Since Y is a h-ideal (left) of X which implies ρY is rough upper h-ideal (left) of X.
$\rho^Y \mathrm{~A}=\rho^Y$, now $\rho^Y S=\rho^Y X S \subseteq \rho^Y A=\rho^Y$.
Theorem 4.8
For any rough upper ideal (right) X of semigroup, if Y is any h-ideal (right) of A, then ρY is rough upper ideal (right).
Proof:
Let $X$ be any ideal (right) of $S$ then $S X \subseteq X$. Also, $Y$, a $h$ ideal (right) of $X$.
$\begin{aligned} & \text { implies } \rho^Y \text { is rough upper } h \text {-ideal(right) of } X, X \rho^Y=\rho_m^Y,S\rho^Y=S X \rho^Y \subseteq X \rho^Y=\rho^Y \text {. }\end{aligned}$
Hence ρY is rough upper ideal (right).
Theorem 4.9
Let $A_1, A_2, A_3, \ldots, A_m$ be any finite set of two-sided ideals of $S . \rho$, a congruence relation on $S$ and $\rho^{A_1}, \rho^{A_2}, \ldots . \rho^{A_m}$ are rough upper ideals (two sided) of $S$ then $\cap_1^m \rho^{A_i} \neq \phi$.
Proof:
Using the method of induction
Given $\rho^{A_1}, \rho^{A_2}, \ldots \ldots \rho^{A_m}$ are rough upper ideals (two sided) of $S$ then
$\rho^{A_1} \rho^{A_2} \subseteq \rho^{A_1} S \subseteq \rho^{A_1}$, also $\rho^{A_1} \rho^{A_2} \subseteq S \rho^{A_2} \subseteq \rho^{A_2}$
$\therefore \rho^{A_1} \rho^{A_2} \subseteq \rho^{A_1} \cap \rho^{A_2}$.
Assume the result is true for m-1 terms.
$\rho^{A_1} \rho^{A_2} \ldots \ldots \rho^{A_{m-1}} \subseteq \bigcap_1^{m-1} \rho^{A_i}$
$\begin{aligned} & \text { Now, } \quad \rho^{A_1} \rho^{A_2} \ldots \ldots \rho^{A_{m-1}} \rho^{A_m} \subseteq \quad \cap_1^{m-1} \rho^{A_i} \cap A_m \subseteq \cap_1^m \rho^{A_i}\end{aligned}$
Since $\rho^{A_1}, \rho^{A_2}, \ldots \ldots \rho^{A_m}$ are non-empty $\therefore \cap_1^m \rho^{A_i} \neq \phi$
Theorem 4.10
Let A1, A2, …., An be any finite set of ideals (left) and $\rho^{A_1}, \rho^{A_2} \ldots .\rho^{A_n}$ are rough upper ideals (left) of S then $\cup_1^m \rho^{A_i}$ is ideal (left) of the semigroup.
Proof:
$\left\{\rho^{A_i}\right\}$, collection of rough upper ideal (left).
Now, $\left(\cup \rho^{A_i}\right) S=\cup\left(\rho^{A_i} S\right)$
Using method of induction
Since $\rho^{A_1} \cup \rho^{A_2}=\rho^{A_1 \cup A_2}$
Also, $\rho^{A_1} S \subseteq \rho^{A_1}, \rho^{A_2} S \subseteq \rho^{A_2} \Rightarrow \rho^{A_1} S \cup \rho^{A_2} S \subseteq \rho^{A_1} \cup$ $\rho^{A_2}$.
Assume the result is true for any m.
$\cup_{i=1}^m \rho^{A_i} S \subseteq \cup_1^m \rho^{A_i}$.
Consider $\cup_{i=1}^{m+1} \rho^{A_i} S=\cup_{i=1}^m \rho^{A_i} S \cup \rho^{A_{m+1}} S \subseteq \cup_1^{m+1} \rho^{A_i}$
Hence for $m+1$ the result is true. Hence by induction, $\cup_1^m \rho^{A_i}$ is an ideal (left) of $S$.
Theorem 4.11
Let $A_1, A_2, \ldots, A_n$ be any finite set of ideals (left) of a semigroup $S$ and $\rho^{A_1}, \rho^{A_2} \ldots . \rho^{A_n}$ are rough upper ideals (left) of $S$ then either $\bigcap_1^n \rho^{A_i}=\phi$ or an ideal (left) of $S$.
Proof:
$\left\{\rho^{A_i}\right\}$ is a collection of rough upper ideal (left) of $S$
Suppose $\cap \rho^{A_i}$ is non empty.
$\begin{aligned} & \cap \rho^{A_i} \supseteq \rho^{\cap A_i} \\ & \rho^{\cap A_i} \subseteq \cap \rho^{A_i} \\ & \rho^{\cap A_i} S \subseteq \cap \rho^{A_i} S \subseteq \cap \rho^{A_i}\end{aligned}$
Theorem 4.12
Let $\rho$ be a congruence relation on $S$. If $A_l$ is a $h$-ideal (left) of $S$ and $B_1$ is an ideal (right) of $S$, then $\rho^{A_1} \subseteq \rho^{B_1}$.
Proof:
Now, $A_1 S=A_1 \Rightarrow A_1 \supseteq A_1 S \& S A_1$
$A_1 B_1 \subseteq A_1 S \subseteq A_1 \& A_1 \subseteq A_1 S \subseteq A_1 B_1$
Then, $A_1 B_1 \subseteq A_1 \& A_1 \subseteq A_1 B_1 \Rightarrow A_1=A_1 B_1$
Also, $A_1 B_1 \subseteq S B_1 \subseteq B_1 \Rightarrow A_1 B_1 \subseteq B_1$
$\rho^{A_1 B_1} \subseteq \rho^{B_1} \Rightarrow \rho^{A_1} \subseteq \rho^{B_1}$
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, …, An 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
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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).
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).
Figure 2. Hasse diagram
Example 5.2
S={1, 2, 3, 4, 5, 6}, along with binary operation * given by
|
* |
1 |
2 |
3 |
4 |
5 |
6 |
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1 |
1 |
1 |
1 |
1 |
1 |
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2 |
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1 |
1 |
1 |
1 |
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3 |
1 |
1 |
1 |
1 |
1 |
1 |
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4 |
1 |
1 |
2 |
1 |
2 |
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5 |
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6 |
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2 |
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).
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$)$.
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 $S$.
Definition 6.1 [18]
$\rho$, a relation(equivalence) on a semigroup. The statements below are therefore implied by each other.
·ρ is congruence
·If $a \rho b$, then $a c \rho b c \,\& \,c a \rho c b, \forall c \in S$
·The collection of all equivalence classes forms a quotient semigroup $S / \rho$ with operation $[\mathrm{a}]_\rho \cdot[\mathrm{b}]_\rho=[\mathrm{ab}]_\rho$
Definition 6.2 [8]
The product of two binary relations $\rho_1 \& \rho_2$ on $S$ is represented by $\rho_1 . P_2$ and is given by $\rho_1 \cdot \rho_2=\{(\alpha, \beta) \in$ $S X S:(\alpha, \gamma) \in \rho_1 \&(\gamma, \beta) \in \rho_2$ for some $\left.\gamma \in S\right\}$
If $\rho_1 \& \rho_2$ are relations (congruence) on $S$, then $\rho_1 . \rho_2$ is also congruence if $\rho_1 . \rho_2=\rho_2 . \rho_1$.
Theorem 6.1 [8]
If $\rho_1 \& \rho_2$ are relations (congruence) on $S$ and if $\rho_{1 . \rho_2=\rho_2 . \rho_1}$ and $A$ be any subsemigroup of $S$ then $\rho_1^A \rho_2^A \subseteq\left(\rho_1 \cdot \rho_2\right)^A$.
Theorem 6.2
If $\tau_1 \& \tau_2$ are congruence relation on $S$ and if $\tau_{1 .} \tau_2=\tau_2 . \tau_l$ and $H$ is any $h$-ideal (left) of $S$ then $\tau_{1_H} \tau_{2_H} \subseteq\left(\tau_1, \tau_2\right)_H$
Proof:
Since H is h-ideal (left) of S then HS=H
Let $\boldsymbol{x} \boldsymbol{y} \in \boldsymbol{\tau}_{\mathbf{1}_{\boldsymbol{H}}} \boldsymbol{\tau}_{\mathbf{2}_{\boldsymbol{H}}}$
$\begin{aligned}&\textbf{x}\in\boldsymbol{\tau}_{\boldsymbol{1_H}}\,\textbf{y}\in\boldsymbol{\tau}_{\mathbf{2_H}}\\&\boldsymbol{{\left[z_1\right]}_{\boldsymbol{\tau_1}} \subseteq \textbf{H},\left[\boldsymbol{z_2}\right]}_{\mathbf{\tau_2}} \subseteq \boldsymbol{H} \,\text {where}\, \boldsymbol{x} \in \boldsymbol{{\left[z_1\right]}_{\mathbf{\tau_1}}, \textbf{y} \in \left[z_2\right]}_{\mathbf{\tau_2}} \\& \mathbf{x} \in \textbf{H}, \mathbf{y} \in \textbf{S} \Rightarrow \mathbf{x y} \in \textbf{H} \\& \mathbf{y} \in \textbf{H}, \mathbf{x} \in \textbf{S} \Rightarrow \mathbf{y x} \in \textbf{H}\end{aligned}$
Also $\boldsymbol{x}\in \mathbf{\tau}_{\mathbf{1_H}} \, \textbf{y} \in \mathbf{\tau}_{\mathbf{2_H}}$
$\begin{aligned}&\Rightarrow (\mathbf{x}, \mathbf{z_1}) \in \mathbf{\tau_1},(\mathbf{y}, \mathbf{z_2}) \in \mathbf{\tau_2} \\& \Rightarrow (\mathbf{x y}, \mathbf{z_1 y}) \in \mathbf{\tau_1},(\mathbf{z_1 y}, \mathbf{z_1 z_2}) \in \mathbf{\tau_2} \\& \Rightarrow (\mathbf{x y}, \mathbf{z_1 z_2}) \in \mathbf{\tau_1} \cdot \mathbf{\tau_2} \\& \Rightarrow \boldsymbol{{\left[z_1 z_2\right]}_{\mathbf{\tau_1} \cdot \mathbf{\tau_2}} \subseteq \textbf{H}} \\& \Rightarrow \mathbf{x y} \in (\mathbf{\tau_1} \cdot \mathbf{\tau_2})_{\textbf{H}}\end{aligned}$
Hence $\mathbf{\tau}_{\mathbf{1_H}} \mathbf{\tau}_{\mathbf{2_H}} \subseteq (\mathbf{\tau_1} \cdot \mathbf{\tau_2})_{\textbf{H}}$
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.
The rough sets theory is crucial to both pure and applied mathematics. In this paper, we focused our study on rough sets algebraic properties as they relate semigroups. Semigroup S is considered as a universe set together with a congruence relation on S forms an approximation space. A connection was made between rough sets and semigroup ideals in this paper. Kuroki [11] introduced the notion of rough upper and lower ideals (left, right, bi ideal). In fact, we considered a h-ideal over a semigroup, and used it to form rough upper and lower h-ideals. Following that, we examined rough h-ideals' properties and results by using suitable examples. As a further extension, we apply rough concepts to semilattices and quotient semigroups The extensive properties of rough semigroup ideals have been discussed. The concept of rough upper and lower semilattices have been illustrated with suitable examples. Further rough quotient semigroup has been discussed. The relative newness of rough set theory is distinguished from the more established ideas. This work may be extended in a similar way to other algebraic structures as well.
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