• Kyungpook Mathematical Journal
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• v.53 no.3
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• pp.435-457
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• 2013

Using the Lukasiewicz 3-valued implication operator, the notion of an (${\alpha},{\beta}$)-intuitionistic fuzzy left (right) $h$-ideal of a hemiring is introduced, where ${\alpha},{\beta}{\in}\{{\in},q,{\in}{\wedge}q,{\in}{\vee}q\}$. We define intuitionistic fuzzy left (right) $h$-ideal with thresholds ($s,t$) of a hemiring R and investigate their various properties. We characterize intuitionistic fuzzy left (right) $h$-ideal with thresholds ($s,t$) and (${\alpha},{\beta}$)-intuitionistic fuzzy left (right) $h$-ideal of a hemiring R by its level sets. We establish that an intuitionistic fuzzy set A of a hemiring R is a (${\in},{\in}$) (or (${\in},{\in}{\vee}q$) or (${\in}{\wedge}q,{\in}$)-intuitionistic fuzzy left (right) $h$-ideal of R if and only if A is an intuitionistic fuzzy left (right) $h$-ideal with thresholds (0, 1) (or (0, 0.5) or (0.5, 1)) of R respectively. It is also shown that A is a (${\in},{\in}$) (or (${\in},{\in}{\vee}q$) or (${\in}{\wedge}q,{\in}$))-intuitionistic fuzzy left (right) $h$-ideal if and only if for any $p{\in}$ (0, 1] (or $p{\in}$ (0, 0.5] or $p{\in}$ (0.5, 1] ), $A_p$ is a fuzzy left (right) $h$-ideal. Finally, we prove that an intuitionistic fuzzy set A of a hemiring R is an intuitionistic fuzzy left (right) $h$-ideal with thresholds ($s,t$) of R if and only if for any $p{\in}(s,t]$, the cut set $A_p$ is a fuzzy left (right) $h$-ideal of R.

• Korean Journal of Mathematics
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• v.10 no.2
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• pp.149-155
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• 2002

We develop some basic properties of $t$-fuzzy ideals in a monoid or a group and find the sufficient conditions for a fuzzy set in a division ring to be a $t$-fuzzy subring and the necessary and sufficient conditions for a fuzzy set in a division ring to be a $t$-fuzzy ideal.

• Honam Mathematical Journal
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• v.32 no.3
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• pp.413-439
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• 2010

W. A. Dudek, M. Shabir and M. Irfan Ali discussed the properties of (${\alpha},{\beta}$)-fuzzy ideals of hemirings in [9]. In this paper, we discuss the generalization of their results on (${\alpha},{\beta}$)-fuzzy ideals of hemirings. As a generalization of the notions of $({\alpha},\;\in{\vee}q)$-fuzzy left (right) ideals, $({\alpha},\;\in{\vee}q)$-fuzzy h-ideals and $({\alpha},\;\in{\vee}q)$-fuzzy k-ideals, the concepts of $({\alpha},\;\in{\vee}q_m)$-fuzzy left (right) ideals, $({\alpha},\;\in{\vee}q_m)$-fuzzy h-ideals and $({\alpha},\;\in{\vee}q_m)$-fuzzy k-ideals are defined, and their characterizations are considered. Using a left (right) ideal (resp. h-ideal, k-ideal), we construct an $({\alpha},\;\in{\vee}q_m)$-fuzzy left (right) ideal (resp. $({\alpha},\;\in{\vee}q_m)$-fuzzy h-ideal, $({\alpha},\;\in{\vee}q_m)$-fuzzy k-ideal). The implication-based fuzzy h-ideals (k-ideals) of a hemiring are considered.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.465-475
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• 2012

We characterize the fuzzy ideal generated by a fuzzy subset in a semigroup and the fuzzy interior ideal generated by a fuzzy subset in a semigroup. Our work generalizes the characterizations ([8]) of those fuzzy ideals generated by fuzzy subsets in semigroups with an identity element.

• East Asian mathematical journal
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• v.22 no.1
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• pp.93-109
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• 2006

The concept of intuitionistic fuzzy set was first introduced by Atanassov in 1986. In this paper, we define the intuitionistic(S,T)-fuzzy left h-ideals of a hemiring by using an s-norm S and a t-norm T and study their properties. In particular, some results of fuzzy left h-ideals in hemirings recently obtained by Jun, $\"{O}zt\"{u}rk$, Song, and others are extended and generalized to intuitionistic (S,T)-fuzzy ideals over hemirings.

• Honam Mathematical Journal
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• v.39 no.4
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• pp.527-559
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• 2017

The main motivation of this article is to generalized the concept of fuzzy ideals, (${\alpha},{\beta}$)-fuzzy ideals, (${\in},{\in}{\vee}q_k$)-fuzzy ideals of semigroups. By using the concept of $q^{\delta}_K$-quasi-coincident of a fuzzy point with a fuzzy set, we introduce the notions of (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy left ideal, (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy right ideal of a semigroup. Special sets, so called $Q^{\delta}_k$-set and $[{\lambda}^{\delta}_k]_t$-set, condition for the $Q^{\delta}_k$-set and $[{\lambda}^{\delta}_k]_t$-set-set to be left (resp. right) ideals are considered. We finally characterize different classes of semigroups (regular, left weakly regular, right weakly regular) in term of (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy left ideal, (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy right ideal and (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy ideal of semigroup S.

• Korean Journal of Mathematics
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• v.19 no.3
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• pp.321-330
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• 2011

We characterize the fuzzy bi-ideal generated by a fuzzy subset in a semigroup and the fuzzy bi-ideal generated by a fuzzy subset A such that $A{\subseteq}A^2$ in a semigroup with an identity element. Our work generalizes the characterization of fuzzy bi-ideals by Mo and Wang ([8]).

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.455-464
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• 2006

In this paper, we introduce the concept of an interval-valued fuzzy left (right, two-sided, interior, bi-) ideal generated by an interval-valued fuzzy subset in semigroups. Some characterizations of such generated interval-valued fuzzy ideals are also discussed.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.551-562
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• 2007

This paper concerns a relationship between rough sets, intuitionistic fuzzy sets and ring theory. We consider a ring as a universal set and we assume that the knowledge about objects is restricted by an intuitionistic fuzzy ideal. We apply the notion of intutionistic fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring. Some properties of the lower and upper approximations are investigated.

• Honam Mathematical Journal
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• v.32 no.4
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• pp.545-561
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• 2010

We introduce the concepts of interval-valued fuzzy sub-groups [resp. normal subgroups, rings and ideals] and investigate some of it's properties.