• International Journal of Fuzzy Logic and Intelligent Systems
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• v.9 no.1
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• pp.53-58
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• 2009

We unite the two con concepts - normality We unite the two con concepts - normality and congruence - in an intuitionistic fuzzy subgroup setting. In particular, we prove that every intuitionistic fuzzy congruence determines an intuitionistic fuzzy subgroup. Conversely, given an intuitionistic fuzzy normal subgroup, we can associate an intuitionistic fuzzy congruence. The association between intuitionistic fuzzy normal sbgroups and intuitionistic fuzzy congruences is bijective and unigue. This leads to a new concept of cosets and a corresponding concept of guotient.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.445-449
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• 2013

We consider a congruence ${\rho}$ on a group G as a subsemigroup of the direct product $G{\times}G$. It is well known that a relation ${\rho}$ on G is a congruence if and only if there exists a normal subgroup N of G such that ${\rho}=\{(s,\;t):st^{-1}{\in}N\}$. In this paper we prove that if G is a finitely presented group, and if N is a normal subgroup of G with finite index, then the congruence ${\rho}=\{(s,\;t):st^{-1}{\in}N\}$ on G is finitely presented.

• Bulletin of the Korean Mathematical Society
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• v.49 no.6
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• pp.1349-1357
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• 2012

We prove a congruence modulo a prime of Fourier coefficients of several meromorphic modular forms of low weights. We prove the result by establishing a generalization of a theorem of Garthwaite.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.13 no.3
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• pp.231-244
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• 2013

We introduce the concept of interval-valued fuzzy congruences on a semigroup S and we obtain some important results: First, for any interval-valued fuzzy congruence $R_e$ on a group G, the interval-valued congruence class Re is an interval-valued fuzzy normal subgroup of G. Second, for any interval-valued fuzzy congruence R on a groupoid S, we show that a binary operation * an S=R is well-defined and also we obtain some results related to additional conditions for S. Also we improve that for any two interval-valued fuzzy congruences R and Q on a semigroup S such that $R{\subset}Q$, there exists a unique semigroup homomorphism g : S/R${\rightarrow}$S/G.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.481-486
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• 2013

In this note, we show that $\mathcal{M}({\Gamma}_0(N))$ is a weighted polynomial ring if and only if N = 1, 2, 4, where $\mathcal{M}({\Gamma}_0(N))$ is the graded ring of integral-weighted modular forms for the congruence subgroup ${\Gamma}_0(N)$.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.345-349
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• 2011

In this work, we find the genera of arithmetic subgroups $\langle{\Gamma}_{\Delta}(N),{\Phi}\rangle$ of $GL_2^+(\mathbb{R})$ generated by congruence subgroup ${\Gamma}_{\Delta}(N)$ and the Fricke involution $\Phi$.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.823-833
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• 2009

It is well-known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of the modular group, then its translates by the group elements form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such hyperbolically convex polygons can be obtained by using Dirichlet's and Ford's polygon constructions. Another method of obtaining a fundamental domain for subgroups of the modular group is through the use of a right coset decomposition and we call such domains standard fundamental domains. In this paper we give subgroups of the modular group which do not have hyperbolically convex standard fundamental domain containing only inequivalent cusps.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.737-748
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• 2009

Using the notion of intuitionistic fuzzy subgroups, its roughness is discussed. With respect to a congruence relation on a group, several properties about the lower and upper approximations of a subset of a group are investigated.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1445-1457
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• 2016

For a positive integer N divisible by 4, let ${\mathcal{O}}^1_N({\mathbb{Q}})$ be the ring of weakly holomorphic modular functions for the congruence subgroup ${\Gamma}^1(N)$ with rational Fourier coefficients. We present explicit generators of the ring ${\mathcal{O}}^1_N({\mathbb{Q}})$ over ${\mathbb{Q}}$ in terms of both Fricke functions and Siegel functions, from which we are able to classify all Fricke families of such level N.

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.565-570
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• 1998

In a number of earlier papers the study of the structure of semigroups has been approached by means of right congruences. Such n approach seems appropriate since a right congruence is one of the possible analogs of both the right ideal of a ring and the subgroup in a group. Each of these substructures plays a strong role in the study of the structure of their respective systems. in both the ring and the group the internal direct product is nat-urally and effectively defined. however what such an internal direct product should be for two right congruences of a semigroup is not so clear. In this paper we will offer a possible definition and consider some of the consequences of it. We will also extend some of these results to automata.