We introduce the notion of fuzzy Abel-Grassmann’s hypergroupoid, hypercongruence, fuzzy hypercongruence, fuzzy strong hypercongruence, compatible relations in an Abel-Grassmann’s hypergroupoid. This ...paper is aimed to study fuzzy hyperideals, smallest fuzzy hyperideals, fuzzy equivalence relations, fuzzy compatible fuzzy strong compatible, fuzzy hypercongruences, fuzzy strong hypercongruences, fuzzy regular, fuzzy strong regular relations and fuzzy hypercongruences in Abel-Grassmann’s hypergroupoids. Characterizations of hypercongruences, their corresponding quotient structure, homomorphisms and an important theorem on embedding Abel-Grassmann’s hypergroupoids by means of fuzzy sets. We show that each hypergroupoid is embedded into a poe-hypergroupoid of all fuzzy subsets of an Abel-Grassmann’s hypergroupoid.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In this paper, we first introduce a new type of rough sets called
α
‐upward fuzzified preference rodownward fuzzy preferenceugh sets using upward fuzy preference relation. Thereafter on the basis of
...α
‐upward fuzzified preference rough sets, we propose approximate precision, rough degree, approximate quality and their mutual relationships. Furthermore, we presented the idea of new types of fuzzy upward
β
‐coverings, fuzzy upward
β
‐neighborhoods and fuzzy upward complement
β
‐neighborhoods and some relavent properties are discussed. Hereby, we formulate a new type of upward lower and upward upper approximations by applying an upward
β
‐neighborhoods. After employing the upward
β
‐neighborhoods based upward rough set approach to it any times, we can only get the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Subsequently, we presented the idea to combine the fuzzy implicator and
t
‐norm to introduce multigranulation
(
ℐ
,
T
)
‐fuzzy upward rough set applying fuzzy upward
β
‐covering and some relative properties are discussed. Finally we presented a new technique for the selection of medicine for treatment of coronavirus disease (COVID‐19) using multigranulation
(
ℐ
,
T
)
‐fuzzy upward rough sets.
In this paper, we prove that the characteristic of a minimal ideal and a minimal generalized ideal, which is meant to be one of minimal left ideal, minimal right ideal, bi-ideal, quasi-ideal, and ...m,n-ideal in a ring, is either zero or a prime number p. When the characteristic is zero, then the minimal ideal (minimal generalized ideal) as additive group is torsion-free, and when the characteristic is p, then every element of its additive group has order p. Furthermore, we give some properties for minimal ideals and for generalized ideals which depend on their characteristics.
In this paper, first, we state an operator LR on an ordered semihyperring R. We show that if φ:R⟶T is a monomorphism and K⊆R, then LT(φ(K))=φ(LR(K)). Afterward, hyperatom elements in ordered ...semihyperrings are defined and some results in this respect are investigated. Denote by A(R) the set of all hyperatoms of R. We prove that if R is a finite ordered semihyperring and |R|≥2, then for any q∈R\{0}, there exists hq∈A*(R)=A(R)\{0} such that hq≤q. Finally, we study the LR-graph of an ordered semihyperring and give some examples. Furthermore, we show that if φ:R⟶T is an isomorphism, G is the LR-graph of R and G′ is the LT-graph of T, then G≅G′.
This paper deals with Krasner hyperrings as an important class of algebraic hyperstructures. We investigate some properties of r-hyperideals in commutative Krasner hyperrings. Some properties of ...pr-hyperideals are also studied. The relation between prime hyperideals and r-hyperideals is investigated. We show that the image and the inverse image of an r-hyperideal are also an r-hyperideal. We also introduce a generalization of r-hyperideals, and we prove some properties of them.
In this paper, we apply the concept of double-framed soft set (briefly DFS-set) to non-associative and non-commutative structure called Abel Grassmann’s groupoid (briefly AG-groupoid). We define ...double-framed soft quasi-ideals (briefly DFS quasi-ideal) of AG-groupoids and discuss their properties in left regular AG-groupoids. We also characterize left regular AG-groupoids in terms of DFS quasi-ideals. Application of DFS sets in decision making situation is provided.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In clasical logic, it is possible to combine the uniary negation operator ¬ with any other binary operator in order to generate the other binary operators. In this paper, we introduce the concept of ...(N∗, O, N, G)-implication derived from non associative structures, overlap function O, grouping function G and two different fuzzy negations N∗ and N are used for the generalization of the implication p → q ≡ ¬ p ∧ ¬ (¬ p ∨ q) . We show that (N∗, O, N, G)-implication are fuzzy implication without any restricted conditions. Further, we also study that some properties of (N∗, O, N, G)-implication that are necessary for the development of this paper. The key contribution of this paper is to introduced the concept of circledcircG,N-compositions on (N∗, O, N, G)-implications. If ( N 1 ∗ , O ( 1 ) , N 1 , G ( 1 ) ) - or ( N 2 ∗ , O ( 2 ) , N 2 , G ( 2 ) ) -implications constructed from the tuples ( N 1 ∗ , O ( 1 ) , N 1 , G ( 1 ) ) or ( N 2 ∗ , O ( 2 ) , N 2 , G ( 2 ) ) satisfy a certain property P, we now investigate whether circledcircG,N-composition of ( N 1 ∗ , O ( 1 ) , N 1 , G ( 1 ) ) - and ( N 2 ∗ , O ( 2 ) , N 2 , G ( 2 ) ) -implications satisfies the same property or not. If not, then we attempt to characterise those implications ( N 1 ∗ , O ( 1 ) , N 1 , G ( 1 ) ) -, ( N 2 ∗ , O ( 2 ) , N 2 , G ( 2 ) ) -implications satisfying the property P such that circledcircG,N-composition of ( M 1 ∗ , O ( 1 ) , M 1 , G ( 1 ) ) - and ( M 2 ∗ , O ( 2 ) , M 2 , G ( 2 ) ) -implications also satisfies the same property. Further, we introduced sup-circledcircO-composition of (N∗, O, N, G)-implications constructed from tuples (N∗, O, N, G) . Subsequently, we show that under which condition sup-circledcircO-composition of (N∗, O, N, G)-implications are fuzzy implication. We also study the intersections between families of fuzzy implications, including RO-implications (residual implication), (G, N)-implications, QL-implications, D-implications and (N∗, O, N, G)-implications.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In this paper, we introduce double-framed soft bi-ideals (briefly DFS bi-ideals) and double-framed soft generalized bi-ideals (briefly DFS generalized bi-ideals) in AG-groupoids and some properties ...of them are investigated. Several characterizations of intra-regular AG-groupoids in terms of DFS left (resp. right) ideals, DFS bi-ideals, DFS generalized bi-ideals and semiprime class of them are provided.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
The notion of cubic set was introduced in 23 as a generalization of the notion of fuzzy sets and intuitionistic fuzzy sets. In this paper, we initiate a study of cubic sets in left almost ...semihypergroups. By using the concept of cubic sets, we introduce the notion of cubic sub LA-semihypergroups (hyperideals and bi-hyperideals) and discuss some basic results on cubic sets in LA-semihypergroups.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
The left invertive law makes Abel Grassmann's groupoids (briefly AG-groupoids) a very interesting structure to study. In this paper, we define (
)-double-framed soft bi-ideals (briefly (
)-DFS ...bi-ideals) and (
)-double-framed soft generalized bi-ideals (briefly (
)-DFS generalized bi-ideals) of AG-groupoids and study some of its properties. We obtain some interesting results of these notions in intra-regular AG-groupoids.
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