Essential Implicative UP-Filters and t -Essential Fuzzy Implicative UP-Filters of UP-Algebras

. For the purpose of this research, we give the new concept of essential implicative UP-ﬁlter from the concept of essential UP-ﬁlters in UP-algebras. Although we know that the concept of implicative UP-ﬁlters is UP-ﬁlters, we have an example showing that the concept of essential UP-ﬁlters does not necessarily be essential implicative UP-ﬁlters. After that, we extend the concept of essential implicative UP-ﬁlters to t -essential fuzzy implicative UP-ﬁlters of UP-algebras and study their relationship based on characteristic functions and upper level subsets.

They are strongly connected with logic.For example, Iséki [13] introduced BCI-algebras in 1966 and has ties with BCI-logic being the BCI-system in combinatory logic, which has application in the language of functional programming.BCK and BCI-algebras are two classes of logical algebras.They were introduced by Imai and Iséki [13,14] in 1966 and have been extensively investigated by many researchers.It is known that the class of BCK-algebras is a proper subclass of the class of BCIalgebras.The concept of KU-algebras was introduced in 2009 by Prabpayak and Leerawat [23].In 2017, Iampan [10] introduced the concept of UP-algebras as a generalization of KU-algebras.
To overcome these uncertainties, researchers are motivated to introduce some classical theories like the theories of fuzzy sets by Zadeh in 1965 [26].Fuzzy set applied in many areas such as medical science, theoretical physics, robotics, computer science, control engineering, information science, measure theory, logic, set theory, topology etc.The study of fuzzy sets is ongoing such as: Al-Masarwah and Ahmad [1] introduced the concept of doubt bipolar fuzzy H-ideals of BCK/BCI-algebras.
Essential fuzzy ideals of rings were studied by Medhi et at in 2008 [16].In 2012, Pawar and Deore introduced the concept of essential ideals in semirings and radical class.They generalized the concept of essential ideals of rings to semirings and established radical class of semirings closed under essential extensions.Later in 2013, Medhi and Saikia [16] studied concept T-fuzzy essential ideals and proved properties of T-fuzzy essential ideals of rings.Later in 2017, Wani and Pawar [18] studied essential ideals and weakly essential ideals in ternary semirings.Amjadi [2] studied the essential ideal graph of a commutative ring in 2018.In 2019, Murugadas et al. [17] studied essential ideals in near-rings using k-quasi coincidence relation.In 2020, Baupradist et at.[3] studied essential ideals and essential fuzzy ideals in semigroups.Together, they studied 0-essential ideals and 0-essential fuzzy ideals in semigroups.In 2021, Gaketem and Iampan [5] introduced the concepts of essential UP-subalgebras and essential UP-ideals and the concepts of t-essential fuzzy UP-subalgebras and t-essential fuzzy UP-ideals of UP-algebras, and investigated their relationships.Moreover, Gaketem et al. [20] studied essential bi-ideals in semigroups.In the same year P. Khamrot and T. Gaketem, [7,8] studied essential ideals in bipolar fuzzy set and interval fuzzy set.Recently, R. Rittichuai et al. [24] disscussed properties of the essential ideals and essential fuzzy ideals in ternary semigroups.
This review shows that the concept of essential subsets is an important and ongoing study, but not much analysis, which has inspired the study of new concepts of essential subsets in UP-algebras.In this paper, we introduce the new concept of essential implicative UP-filters from the concept of essential UP-filters in UP-algebras and study some properties.We will show that the concept of essential UP-filters does not necessarily be essential implicative UP-filters.Finally, we extend the concept of fuzzy implicative UP-filters to t-essential fuzzy implicative UP-filters of UP-algebras and study their relationship based on characteristic functions and upper level subsets.

Preliminaries
Now, we discuss the concept of UP-algebras and basic properties for the study of next sections.Definition 2.1.[10] An algebra A := (A, •, 0) of type (2, 0) is called a UP-algebra, where A is a nonempty set, • is a binary operation on A, and 0 is a fixed element of A (i.e., a nullary operation) if it satisfies the following axioms: From [10], we know that the concept of UP-algebras is a generalization of KU-algebras.
The binary relation ≤ on a UP-algebra A is defined as follows: and the following assertions are valid (see [10,11]). (for ) (for all a, x, y , z (2.18) Example 2.1.[21] Let U be a nonempty set and let X ∈ P(U) where P(U) means the power set of U. Let P X (U) = {A ∈ P(U) | X ⊆ A}.Define a binary operation on P X (U) by putting where A C means the complement of a subset A. Then (P X (U), , X) is a UP-algebra.Let P X (U) = {A ∈ P(U) | A ⊆ X}.Define a binary operation on (3) an implicative UP-filter of A if (a) the constant 0 of A is in S, and Every implicative UP-filter is a UP-filter, but the converse is not true in general as study in the following example.
For any two fuzzy sets ω and in a nonempty set S, we define (2) ω = ⇔ ω ≥ and ≥ ω. ( If K ⊆ S, then the characteristic function ω K of S is a function from S into {0, 1} defined as follows: We easily prove that if ω 1 and ω 2 are fuzzy implicative UP-filters of a UP-algebra A, then ω 1 ∧ ω 2 is also a fuzzy implicative UP-filter of A.
Lemma 2.1.[22] Let F be a nonempty subset A. Then the constant 0 of A is in F if and only if Theorem 2.1.Let F be a nonempty subset of a UP-algebra A. Then F is an implicative UP-filter of A if and only if the characteristic function ω F is a fuzzy implicative UP-filter of A.
Proof.Assume that F is an implicative UP-filter of A. Since 0 ∈ F , it follows from Lemma 2.1 that From cases 1 and 2, we have that ω F is a fuzzy implicative UP-filter of A.
Conversely, assume that ω F is a fuzzy implicative UP-filter of A. Since ω F (0) ≥ ω F (x) for all x ∈ A, it follows form Lemma 2.1 that 0 ∈ F .Let x, y , z ∈ A be such that x It is contradiction.So, x • z ∈ A. Hence F is an implicative UP-filter of A.
Definition 2.5.[22] Let ω be a fuzzy set in a UP-algebra A. For any t ∈ [0, 1], the sets are called an upper t-level subset and an upper t-strong level subset of ω, respectively.
Theorem 2.2.Let ω be a fuzzy set in a UP-algebra A. Then the following statements hold: (1) ω is a fuzzy implicative UP-filter of A if and only if for any t ∈ [0, 1], U(ω; t) is an implicative ω is a fuzzy implicative UP-filter of A if and only if for any t ∈ [0, 1], U + (ω; t) is an implicative Proof.
By assumption, we have U(ω; t) is an implicative UP-filter of A and so x • z ∈ U(ω; t).Thus Hence ω is a fuzzy implicative UP-filter of A.
3. Essential implicative UP-filters and t-essential fuzzy implicative UP-filters In this section, we define essential implicative UP-filters and essential fuzzy implicative UP-filters, together study their basic properties.
First, let us recall the definition of an essential UP-filter of a UP-algebra as follows: is a nonzero subset (actually, it is a nonzero UP-filter) of A for every nonzero UP-filter C of A.
Equivalently, {0} ⊂ B ∩ C for every nonzero UP-filter C of A.
In the same way, we define an essential implicative UP-filter of a UP-algebra as follows: Example 3.1.By Example 2.2 we have {0, 2, 3} and A are essential UP-filters, and A is the only one essential implicative UP-filter of A. This example shows that an essential implicative UP-filter is not necessarily an essential UP-filter.
Theorem 3.1.Let F be an essential implicative UP-filter of A and F be a implicative UP-filter of A containing F .Then F is also an essential implicative UP-filter of A.
Proof.Let G be a nonzero implicative UP-filter of A. Since F is an essential implicative UP-filter of Hence, F is an essential implicative UP-filter of A.
Theorem 3.2.Let F 1 and F 2 be an essential implicative UP-filters of A Then F 1 ∩ F 2 is also an essential implicative UP-filter of A.
Proof.Let C be a nonzero implicative UP-filter of A. Since F 1 and F 2 are implicative UP-filters of A, ).A fuzzy implicative UP-filter ω of A is called a t-essential fuzzy implicative UP-filter of A if there exists a nonzero element x ∈ A such that t < (ω ∧ )(x ) for every nonzero fuzzy implicative UP-filter of A.
Theorem 3.3.Let ω be a t-essential fuzzy implicative UP-filter of A and ω be a fuzzy implicative UP-filter of A such that ω ≤ ω .Then ω is also a t-essential fuzzy implicative UP-filter of A.
Proof.Let be a nonzero fuzzy implicative UP-filter of A. Since ω is a t-essential fuzzy implicative UP-filter of A, there exists a nonzero element x ∈ A such that t < (ω ∧ )(x ) ≤ (ω ∧ )(x ).
Hence, ω is a t-essential fuzzy implicative UP-filter of A.
Proof.Let be a fuzzy set in A defined by (x) = 1 for all x ∈ A. Then we can easily prove that is a nonzero fuzzy implicative UP-filter of A. Then there exists a nonzero element x ∈ A such that t < (ω ∧ )(x ) = min{ω(x ), (x )} = min{ω(x ), 1} = ω(x ) ≤ ω(0).
Proof.Let be a nonzero fuzzy implicative UP-filter of A. Since ω 1 and ω 2 are fuzzy implicative UP-filters of A, we have ω 1 ∧ ω 2 is a fuzzy implicative UP-filter of A.
Since ω 2 is a t-essential fuzzy implicative UP-filter of A, we have ω 2 ∧ is a nonzero fuzzy implicative UP-filter of A. Since ω 1 is a t-essential fuzzy implicative UP-filter of A, there exists a nonzero element fuzzy implicative UP-filter of A.
Theorem 3.6.If F is an essential implicative UP-filter of a nonzero UP-algebra A, then characteristic function ω F is a 0-essential fuzzy implicative UP-filters of A.
Proof.Suppose that F is an essential implicative UP-filter of A. Then F is an implicative UP-filter of A. Thus ω F is a fuzzy implicative UP-filter of A. Let be a nonzero fuzzy implicative UP-filter of A.
Theorem 3.7.Let F be a UP-filter of a nonzero UP-algebra A. If the characteristic function ω F is a t-essential fuzzy implicative UP-filter of A, then F is an essential implicative UP-filter of A.
Proof.Assume that ω F is a t-essential fuzzy implicative UP-filter of A. Then ω F is a fuzzy implicative UP-filter of A. Thus by Theorem 2.1, we have that F is an implicative UP-filter of A. Let F be a nonzero UP-filter of A. Thus by Theorem 2.1, we have ω F is a nonzero fuzzy implicative UP-filter of A. Since ω F is a t-essential fuzzy implicative UP-filter of A, there exists a nonzero element x ω F ∈ A such that t < (ω F ∧ ω F )(x ω F ) = min{ω F (x ω F ), ω F (x ω F )}.This implies that x ω F ∈ F ∩ F , that is, {0} ⊂ F ∩ F .Hence, F is an essential implicative UP-filter of A.
a UP-filter of A if (a) the constant 0 of A is in S, and (b) (for all x, y ∈ A)(x ∈ S, x • y ∈ S ⇒ y ∈ S).