DESCRIBING GREEN'S RELATIONS IN ORDERED Γ-GROUPOIDS USING A NEW CONCEPT: FUZZY SUBSETS - Iampan, Aiyared; Siripitukdet, Manoj (Italian Journal of Pure and Applied Mathematics, 2013)
This work extends the idea of the Green's relations \(\mathcal{R}\), \(\mathcal{L}\), \(\mathcal{I}\) and the usual relation \(\mathcal{N}\) of ordered groupoids to ordered Γ-groupoids in terms of fuzzy subsets, the concept of the Green's relations \(\mathcal{R}\), \(\mathcal{L}\), \(\mathcal{I}\) and the usual relation \(\mathcal{N}\) of ordered Γ-groupoids is introduced and investigated, which is an interesting for ordered Γ-groupoids and some interesting characterizations of the Green's relations \(\mathcal{R}\), \(\mathcal{L}\), \(\mathcal{I}\) and the usual relation \(\mathcal{N}\) of ordered Γ-groupoids are obtained. Also, we define relations \(\mathcal{R}\)F, \(\mathcal{L}\)F, \(\mathcal{I}\)F and \(\mathcal{N}\)F of ordered Γ-groupoids in terms of fuzzy subsets and we prove that \(\mathcal{R}\)F = \(\mathcal{R}\), \(\mathcal{L}\)F = \(\mathcal{L}\), \(\mathcal{I}\)F = \(\mathcal{I}\) and \(\mathcal{N}\)F = \(\mathcal{N}\) in the same way as of the results of Kehayopulu and Tsingelis [8].