• Title, Summary, Keyword: congruence subgroup

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SEMIGROUP PRESENTATIONS FOR CONGRUENCES ON GROUPS

  • Ayik, Gonca;Caliskan, Basri
    • 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.

INTUITIONISTIC FUZZY NORMAL SUBGROUP AND INTUITIONISTIC FUZZY ⊙-CONGRUENCES

  • Hur, Kul;Kim, So-Ra;Lim, Pyung-Ki
    • 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.

Interval-Valued Fuzzy Congruences on a Semigroup

  • Lee, Jeong Gon;Hur, Kul;Lim, Pyung Ki
    • 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.

ESTIMATION OF GENUS FOR CERTAIN ARITHMETIC GROUP

  • Jeon, Daeyeol
    • 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 $\Phi$.

ROUGHNESS BASED ON INTUITIONISTIC FUZZY SUBGROUPS

  • Baik, Hyoung-Gu;Jun, Young-Bae;Park, Chul-Hwan
    • 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.

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NONEXISTENCE OF H-CONVEX CUSPIDAL STANDARD FUNDAMENTAL DOMAIN

  • Yayenie, Omer
    • 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.

DETERMINATION OF THE FRICKE FAMILIES

  • Eum, Ick Sun;Shin, Dong Hwa
    • 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.