Here is a 3-paragraph summary of the blog post on complete mappings of semigroups:
A magma is simply a set equipped with a binary operation, and a complete mapping is a special kind of bijection defined on such a structure. Specifically, a bijection θ on a magma A is called a complete mapping if the function ψ defined by ψ(a) = a · θ(a) is also a bijection. This elegant condition links two bijections together through the magma's operation, making complete mappings a rich and interesting object of study.
Complete mappings have connections to many areas of combinatorics and algebra, including the construction of Latin squares and the study of groups. Not every algebraic structure admits a complete mapping, and determining which ones do is a central question in this area of research. For groups, a classical theorem states that a finite group has a complete mapping if and only if its Sylow 2-subgroup is trivial or non-cyclic, giving a clean characterization tied to the group's structure.
Extending these ideas to semigroups, which are magmas where the binary operation is associative but need not have an identity or inverses, presents new challenges and open questions. The behavior of complete mappings becomes more complex and varied in this broader setting, making it fertile ground for mathematical exploration. Understanding when semigroups admit complete mappings deepens our knowledge of how algebraic structure constrains the mappings a set can support.