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Homotopy theory is one of the main areas of algebraic topology. In [7] the concept of H-spaces (Hopf spaces) is introduced from the viewpoint of homotopy theory, then a grouplike space which is a group up to homotopy, called an H-group (Hopf group) is defined. An H-space is a pointed space (𝑋, 𝑥₀) with a constant map 𝑐: 𝑋→ 𝑋, 𝑥→ 𝑥₀, for all 𝑥∈ 𝑋 and a continuous multiplication 𝑚: 𝑋× 𝑋→ 𝑋 such that 𝑥₀ is a homotopy identity, that is, the diagram 𝑋 (𝑐, 1𝑋)→ 𝑋× 𝑋 (1𝑋, 𝑐)→ 𝑋 1𝑋 𝑚 1𝑋 commutes up to homotopy: 𝑚∘(𝑐, 1𝑋)≃ 1𝑋≃ 𝑚∘(1𝑋, 𝑐). Also if an H-space (𝑋, 𝑥₀) has a homotopy commutative multiplication 𝑚 (𝑖. 𝑒. 𝑚∘(𝑚× 1𝑋)≃ 𝑚∘(1𝑋× 𝑚)) and has a homotopy invers 𝑛 (ie there exist a function 𝑛: 𝑋→ 𝑋 such that 𝑚∘(𝑛, 1𝑋)≃ 𝑐≃ 𝑚∘(1𝑋, 𝑛)), then (𝑋, 𝑥₀) is called a H-group. If there exist a function 𝑇: 𝑋× 𝑋→ 𝑋× 𝑋, 𝑇 (𝑥, 𝑦)=(𝑦, 𝑥) such that 𝑚∘ 𝑇≃ 𝑚, then H-space (𝑋, 𝑥₀) is called an abelian Hspace. In [13] Zadeh introduced the concepts of fuzzy sets. After Chang developed the theory of fuzzy topological spaces [3], basic concepts from homotopy theory were discussed in fuzzy settings. In this direction, Zheng [4] introduced the concept of fuzzy paths. Also, in [2], fuzzy homotopy concepts in fuzzy topological spaces were conceived. In this study firstly, I define fuzzy H-space and fuzzy H-group. Then I show that a fuzzy loop space ΩX is a fuzzy H-group with the continuous multiplication 𝑚: 𝛺𝑋× 𝛺𝑋→ 𝛺𝑋, 𝑚 (𝛼 (𝐸), 𝛽 (𝐷))= 𝛾 (𝐸+ 𝐷). Then I show that a fuzzy retract of a fuzzy loop space is a fuzzy H-space. Besides, a fuzzy deformation retract is defined and it is shown that a fuzzy deformation retract of a fuzzy loop space is a fuzzy H-group …

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On Fuzzy Retract of a Fuzzy Loop Space

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