Multi-adjoint algebras versus non-commutative residuated structures

•Multi-adjoint algebras and several properties have been introduced.•Adjoint triple and its “dual” cannot be considered in the same framework.•A comparison among general residuated algebraic structures has been introduced.•Multi-adjoint algebras generalize these structures with residuated implications.

Adjoint triples and pairs are basic operators used in several domains, since they increase the flexibility in the framework in which they are considered. This paper introduces multi-adjoint algebras and several properties; also, we will show that an adjoint triple and its “dual” cannot be considered in the same framework.Moreover, a comparison among general algebraic structures used in different frameworks, which reduce the considered mathematical requirements, such as the implicative extended-order algebras, implicative structures, the residuated algebras given by sup-preserving aggregations and the conjunctive algebras given by semi-uninorms and u-norms, is presented. This comparison shows that multi-adjoint algebras generalize these structures in domains which require residuated implications, such as in formal concept analysis, fuzzy rough sets, fuzzy relation equations and fuzzy logic.