# Analysis of Topological Endomorphism of Fuzzy Measure in Hausdorff Distributed Monoid Spaces

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Definitions

#### 3.1. Generator Order

#### 3.2. Endomorphic Topological Map

#### 3.3. Cyclic Map

#### 3.4. Cyclic Topological Endomorphism

## 4. Analytical Properties

**Theorem**

**1.**

**Proof:**

**Theorem**

**2.**

**Proof:**

**Theorem**

**3.**

**Proof:**

**Theorem**

**4.**

**Proof:**

**Theorem**

**5.**

**Proof:**

**Theorem**

**6.**

**Proof:**

**Lemma**

**1.**

**Proof:**

**Theorem**

**7.**

**Proof:**

**Theorem**

**8.**

**Proof:**

**Corollary**

**1.**

**Theorem**

**9.**

**Proof:**

**Theorem**

**10.**

**Proof:**

## 5. Comparative Evaluation

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Comparison of models based on group structures and topologies. EFM: fuzzy topological measure in cyclic endomorphic distributed monoid (DM) space; EMG: endomorphic measure on groups; MIS: measure-invariant semigroups; PML: probability-measured Lie groups; TAG: topological actions on groups.

Group and Topological Properties | Compactness/Finiteness | Commutativity | Topology | Embeddings |
---|---|---|---|---|

Model: TAG | Finite | Non-Abelian | Orientation-preserving | Fixed point |

Model: MIS | Locally compact | Abelian | Translation-invariant | Not applicable |

Model: PML | Borel sets | Not applicable | Weak convergent | Not applicable |

Model: EMG | Amenable locally compact | Polish groups | Second-countable | Not applicable |

Model: EFM | Finite, compact | Not applicable | Second-countable | Monoids |

Measure Properties | Translation | Symmetry | Isomorphism/Endomorphism | Finiteness | Haar Condition | Measure |
---|---|---|---|---|---|---|

Model: TAG | Orientation-preserving | No | Isomorphic | Not applicable | Not applicable | Not applicable |

Model: MIS | Linear additive | No | Translation-invariant | Finite | Not applicable | Non-fuzzy |

Model: PML | Not applicable | Yes | Not applicable | Yes | Yes | Borel measure |

Model: EMG | Measure-preserving | Not applicable | Automorphic | Sigma-finite | Yes | Probability-preserving |

Model: EFM | Linear or non-linear | Depends on translation | Endomorphic | Yes | Yes | Fuzzy measure |

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Bagchi, S.
Analysis of Topological Endomorphism of Fuzzy Measure in Hausdorff Distributed Monoid Spaces. *Symmetry* **2019**, *11*, 671.
https://doi.org/10.3390/sym11050671

**AMA Style**

Bagchi S.
Analysis of Topological Endomorphism of Fuzzy Measure in Hausdorff Distributed Monoid Spaces. *Symmetry*. 2019; 11(5):671.
https://doi.org/10.3390/sym11050671

**Chicago/Turabian Style**

Bagchi, Susmit.
2019. "Analysis of Topological Endomorphism of Fuzzy Measure in Hausdorff Distributed Monoid Spaces" *Symmetry* 11, no. 5: 671.
https://doi.org/10.3390/sym11050671