  This paper deals with the decidability of semigroup freeness. More precisely,
the freeness problem over a semigroup S is defined as: given a finite subset X
of S, decide whether each element of S has at most one factorization over X. To
date, the decidabilities of two freeness problems have been closely examined.
In 1953, Sardinas and Patterson proposed a now famous algorithm for the
freeness problem over the free monoid. In 1991, Klarner, Birget and Satterfield
proved the undecidability of the freeness problem over three-by-three integer
matrices. Both results led to the publication of many subsequent papers. The
aim of the present paper is three-fold: (i) to present general results
concerning freeness problems, (ii) to study the decidability of freeness
problems over various particular semigroups (special attention is devoted to
multiplicative matrix semigroups), and (iii) to propose precise, challenging
open questions in order to promote the study of the topic.
