• Honam Mathematical Journal
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• v.38 no.4
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• pp.795-807
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• 2016

More general form of an (${\in},{\in}{\vee}q_k$)-fuzzy subsemigroup is considered. The notions of (${\in},q^{\delta}_k$)-fuzzy subsemigroup, ($q^{\delta}_0,{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup and (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup are introduced, and related properties are investigated. Characterizations of an (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup are considered. Conditions for an (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup to be a fuzzy subsemigroup are provided. Relations between ($q^{\delta}_0,{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup, (${\in},q^{\delta}_k$)-fuzzy subsemigroup and (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy subsemigroup are discussed.

• 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.

• Honam Mathematical Journal
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• v.39 no.2
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• pp.199-216
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• 2017

In the paper [Y. B. Jun, B. Davvaz and A. Khan, Filters of ordered semigroups based on the fuzzy points, JIFS 24 (2013) 619-630]. Jun et al. discussed the notion of (${\in},{\in}{\vee}q_k$)-fuzzy left (resp., right) filters as a generalization of the notion of (${\in},{\in}{\vee}q$)-fuzzy left (resp., right) filters of ordered semigroups. In this article, we try to obtain a more general form that (${\in},{\in}{\vee}q_k$)-fuzzy left (resp., right) filters in ordered semigroups. The notion of (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp., right) filters is discussed, and several properties are investigated. Characterizations of an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp., right) filter are established. A condition for an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp., right) filter to be a fuzzy left (resp., right) filter is provided. The important achievement of the study with an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (right) filter is that the notion of an (${\in},{\in}{\vee}q_k$)-fuzzy left ( right) filter and hence an (${\in},{\in}{\vee}q$)-fuzzy left (resp. right) filter are special cases of an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp. right) filter, and thus several results in published papers are becoming corollaries of our results obtained in this paper.

• East Asian mathematical journal
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• v.31 no.1
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• pp.1-18
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• 2015

In this paper we characterize different classes of hemirings by the properties of their (${\in},{\in}{\vee}q$)-fuzzy ideals, (${\in},{\in}{\vee}q$)-fuzzy quasi-ideals and (${\in},{\in}{\vee}q$)-fuzzy bi-ideals.

• Bulletin of the Korean Mathematical Society
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• v.41 no.4
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• pp.723-730
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• 2004

The definition of a D-admissible fuzzy subset for an operator domain D on a group G is modified to obtain new kinds of (${\in},\;{\in}\;{\vee}q$)-fuzzy subgroups such as an (${\in},\;{\in}\;{\vee}q$)-fuzzy normal subgroup, an (<${\in},\;{\in}\;{\vee}q$)-fuzzy characteristic subgroup, an (<${\in},\;{\in}\;{\vee}q$)-fuzzy fully invariant subgroup which are invariant under D. As results, some of the fundamental properties of such (${\in},\;{\in}\;{\vee}q$)-fuzzy subgroups are obtained.

• Honam Mathematical Journal
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• v.41 no.3
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• pp.581-593
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• 2019

The notions of (${\in}$, ${\in}{\vee}q$)-permeable S-value and (${\in}$, ${\in}{\vee}q$)-permeable I-value are introduced, and related properties are investigated. Relations among (${\in}$, ${\in}{\vee}q$)-fuzzy subalgebra, (${\in}$, ${\in}{\vee}q$)-fuzzy ideal, (strong) lower and (strong) upper level sets, (${\in}$, ${\in}{\vee}q$)-permeable S-value, (${\in}$, ${\in}{\vee}q$)-permeable I-value, S-energetic set, I-energetic set, right stable set and right vanished set are discussed.

• 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.

• Honam Mathematical Journal
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• v.35 no.4
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• pp.583-606
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• 2013

We generalize the idea of (${\in}$, ${\in}{\vee}q_k$)-fuzzy ordered semi-group and give the concept of (${\in}$, ${\in}{\vee}q_k$)-fuzzy ordered $\mathcal{LA}$-semigroup. We show that (${\in}$, ${\in}{\vee}q_k$)-fuzzy left (right, two-sided) ideals, (${\in}$, ${\in}{\vee}q_k$)-fuzzy (generalized) bi-ideals, (${\in}$, ${\in}{\vee}q_k$)-fuzzy interior ideals and (${\in}$, ${\in}{\vee}q_k$)-fuzzy (1, 2)-ideals need not to be coincide in an ordered $\mathcal{LA}$-semigroup but on the other hand, we prove that all these (${\in}$, ${\in}{\vee}q_k$)-fuzzy ideals coincide in a left regular class of an ordered $\mathcal{LA}$-semigroup. Further we investigate some useful conditions for an ordered $\mathcal{LA}$-semigroup to become a left regular ordered $\mathcal{LA}$-semigroup and characterize a left regular ordered $\mathcal{LA}$-semigroup in terms of (${\in}$, ${\in}{\vee}q_k$)-fuzzy one-sided ideals. Finally we connect an ideal theory with an (${\in}$, ${\in}{\vee}q_k$)-fuzzy ideal theory by using the notions of duo and (${\in}{\vee}q_k$)-fuzzy duo.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.431-439
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• 2012

Characterizations of $\mathcal{N}$-subalgebra of type (${\in}$, ${\in}{\vee}q$) are provided. The notion of $\mathcal{N}$-subalgebras of type ($\bar{\in}$, $\bar{\in}{\vee}\bar{q}$) is introduced, and its characterizations are discussed. Conditions for an $\mathcal{N}$-subalgebra of type (${\in}$, ${\in}{\vee}q$) (resp. ($\bar{\in}$, $\bar{\in}{\vee}\bar{q}$) to be an $\mathcal{N}$-subalgebra of type (${\in}$, ${\in}$) are considered.

• East Asian mathematical journal
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• v.27 no.3
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• pp.349-372
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• 2011

In this paper we define interval valued (${\in}$, ${\in}{\vee}q$)-fuzzy hquasi-ideals, interval valued (${\in}$, ${\in}{\vee}q$)-fuzzy h-bi-ideals, interval valued ($\bar{\in}$, $\bar{\in}{\vee}\bar{q}$)-fuzzy h-ideals, interval valued ($\bar{\in}$, $\bar{\in}{\vee}\bar{q}$)-fuzzy h-quasi-ideals, interval valued ($\bar{\in}$, $\bar{\in}{\vee}\bar{q}$)-fuzzy h-bi-ideals and characterize different classes of hemirings by the properties of these ideals.