Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585318120230201The prime state ideal theorem in state residuated lattices13115310328810.52547/cgasa.2023.103288ENFrancis WoumfoDepartement of Mathematics and Computer Science, Faculty of Sciences, University of Dschang, Dschang, Cameroon0000-0002-0479-4309Etienne Romuald AlomoTemgouaDepartment of Mathematics, Ecole Normale Sup´erieure, de Yaound´e, University of Yaound´e 1, P.O. Box 47, Yaound´e, Cameroon.0000-0001-6338-090XCelestin LeleDepartment of Mathematics and Computer Science, Faculty of Science, University of Dschang, Dschang, Cameroon0000-0001-6338-090XJournal Article20230313The aim of this paper is to establish the prime state ideal theorem in state residuated lattices (SRLs). We study the state ideals lattice $\mathcal{SI}(L)$ of a state residuated lattice $(L, \varphi)$ and prove that it is a complete Brouwerian lattice in which the meet and the join of any two compact elements are compact (coherent frame). We characterize the notion of prime state ideals in SRLs. In addition, we establish the condition for which the lattice $\mathcal{SI}(L)$ is a Boolean algebra. https://cgasa.sbu.ac.ir/article_103288_7c80a9cedf8e1ba5002830b5d68d31cf.pdf