Hyperrectangular and hyperrhombic functional equations

Abstract

The classical rectangular and rhombic functional equations for functions mapping \({\mathbb {R}}\times {\mathbb {R}}\) into \({\mathbb {R}}\) were solved in an article (Aequat. Math. 1, 37-53, 1968) by Aczél, H. Haruki, McKiernan and Sakovič about 60 years ago. The authors showed that the two equations are equivalent and their solutions of are the sum of a bi-additive function, additive functions of each variable, and a constant. A recent article (Aequat. Math. (2026) 100:69) by Lin, Yang and Zhao extended the results on the rectangular equation to hyperrectangular equations on groups. In the present article we generalize those results by solving the hyperrectangular and hyperrhombic functional equations on monoids with an involution.