Malaysian Journal of Mathematical Sciences, May 2021, Vol. 15, No. 2
A New View of Intra-Regular \(\mathcal{AG}\)-Groupoids in Terms of Generalized Cubic Ideals
Gulistan, M., Kadry, S., and Azhar, M.
Corresponding Email: s.kadry@bau.edu.lb
Received date: 11 November 2019
Accepted date: 20 June 2020
Abstract:
In this paper we characterize the intra-regular \(\mathcal{AG}\)-groupoids in terms of generalized cubic set. We show that the concept of \((\in_{\Gamma}, \in_{\Gamma}\vee q\Delta)\)-cubic ideals and of \((\in_{\Gamma}, \in_{\Gamma}\vee q\Delta)\)-cubic interior ideals in an intra-regular \(\mathcal{AG}\)-groupoid \(S\) with left identity coincides. We additionally demonstrate that an \(\mathcal{AG}\)-groupoid \(S\) with left identity is intra-regular if and only if \(\beta_1\wedge^{\ast} \beta_2=\beta_1\circ ^{\ast} \beta_2\) for all \((\in_{\Gamma}, \in_{\Gamma}\vee q\Delta)\)-cubic quasi ideals \(\beta_1=\left \langle \widetilde{\Im}_{\beta_1}, \hbar_{\beta_1} \right \rangle\) and \(\beta_2=\left \langle \widetilde{\Im}_{\beta_2}, \hbar_{\beta_2} \right \rangle\) of \(S\).
Keywords: \(\mathcal{AG}\)-groupoids, intra-regular \(\mathcal{AG}\)-groupoids, cubic sets, generalized sub \(\mathcal{AG}\)-groupoids, generalized cubic ideals
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