On local triangle algebras

In this paper, we have introduced the notion of local and semilocal triangle algebras and propose the theorems that characterize these algebraic structures. Additionally, we have established the new properties of these algebraic structures and discussed the relations between local triangle algebras and some interval valued residuated lattice (IVRL)-filters, such as n-fold IVRL-extended integral filters and IVRL-extended maximal filters. The obtained results proved that the MTL-triangle algebra is a subdirect product of local triangle algebras. Moreover, a correlation was observed between the set of the dense elements and local triangle algebras. Finally, semilocal triangle algebras were introduced and assessed in detail, and an association was observed between the semilocal triangle algebras and quotient triangle algebras.