mathematics Article Graphs Based on Hoop Algebras Mona Aaly Kologani 1, Rajab Ali Borzooei 2 and Hee Sik Kim 3,* 1 Hatef Higher Education Institute, Zahedan 8301, Iran;
[email protected] 2 Department of Mathematics, Shahid Beheshti University, Tehran 7561, Iran;
[email protected] 3 Research Institute for Natural Science, Department of Mathematics, Hanyang University, Seoul 04763, Korea * Correspondence:
[email protected]; Tel.: +82-2-2220-0897 Received: 14 March 2019; Accepted: 17 April 2019 Published: 21 April 2019 Abstract: In this paper, we investigate the graph structures on hoop algebras. First, by using the quasi-filters and r-prime (one-prime) filters, we construct an implicative graph and show that it is connected and under which conditions it is a star or tree. By using zero divisor elements, we construct a productive graph and prove that it is connected and both complete and a tree under some conditions. Keywords: hoop algebra; zero divisor; implicative graph; productive graph MSC: 06F99; 03G25; 18B35; 22A26 1. Introduction Non-classical logic has become a formal and useful tool for computer science to deal with uncertain information and fuzzy information. The algebraic counterparts of some non-classical logics satisfy residuation and those logics can be considered in a frame of residuated lattices [1]. For example, Hájek’s BL (basic logixc [2]), Lukasiewicz’s MV (many-valued logic [3]) and MTL (monoidal t-norm-based logic [4]) are determined by the class of BL-algebras, MV-algebras and MTL-algebras, respectively. All of these algebras have lattices with residuation as a common support set.