On Hom-Analogues of Heaps and Trusses

Abstract: This paper develops the foundations of Hom-heaps, Hom-trusses, and Hom-braces as natural Hom-type analogues of their classical counterparts. We establish the correspondence between Hom-heaps and Hom-groups, showing that the retract of a Hom-heap at a point yields a Hom-group precisely when the point is fixed under the twisting map. Three interrelated definitions of Hom-trusses are introduced and their structural properties are investigated, including the construction of modules over Hom-trusses. For Hom-braces, we propose three variants and demonstrate their close connections with Hom-trusses, proving in particular that certain Hom-trusses naturally give rise to Hom-braces and vice versa. Our results provide a systematic framework that extends heap and truss theory into the Hom-algebraic setting, opening new perspectives for applications in Yang--Baxter theory, non-associative geometry, and categorical algebra.