DESCRIBING GREEN'S RELATIONS IN ORDERED Γ-GROUPOIDS USING A NEW CONCEPT: FUZZY SUBSETS

Abstract

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].