Complete Mappings of Semigroups

Abstract

A complete mapping of a semigroup $S$ is a bijection $\alpha\colon S\to S$ such that the map $\theta\colon S\to S$ defined by $x\theta=x\cdot x\alpha$ is also a bijection. Equivalently, it determines a transversal of the multiplication table of $S$. Complete mappings connect group theory, Latin squares, and cryptography, and their existence for finite groups was characterized by the resolution of the Hall--Paige conjecture. In this paper, we develop the corresponding theory for finite semigroups. We prove that every finite semigroup admitting a complete mapping is regular and that the problem reduces to principal factors. We classify the existence of a complete mapping in Rees matrix semigroups without zero, give a Hall-type criterion for Rees $0$-matrix semigroups over groups with complete mappings, and prove sufficient conditions for Rees $0$-matrix semigroups whose maximal subgroups do not have complete mappings. As the main application of the Rees $0$-matrix analysis, we show that $T_n$ has a complete mapping if and only if $n=1$ or $n\geq 4$. Equivalently, $T_n$ has a complete mapping if and only if the same holds for $S_n$. We prove that the full linear monoid of a finite-dimensional vector space has a complete mapping except in dimension $1$ over a field of odd order and in dimension $2$ over $\mathbb F_2$. We also prove that the partition monoid $\mathcal P_n$ has a complete mapping if and only if $n=1$ or $n\ge4$, and that every finite aperiodic regular $*$-semigroup has a complete mapping. As a consequence, the planar partition, Motzkin and Jones monoids have complete mappings. The paper concludes with open problems.