Quasi-idempotents in certain transformation semigroups

Let S be a semigroup and e∈S. If e≠e^2=e^4, then e∈S is called a quasi-idempotent.. This concept corresponds to the concept of involution in groups. Although there appears to be almost nothing that can be said about the structure of a subgroup generated by two elements of given orders m≥1 and n≥1 in any case other than m = n = 2, two quasi-idempotent in any group generate a dihedral subgroup. Moreover quasi-idempotents also have an important role for group presentation, since there is no need to use the inverse of any generators in relations since the inverse of any involution is itself. Hence determining the structure of quasi-idempotents in certain groups and semigroups is an important research topic. Let P_n, T_(n,) 〖 I〗_n, S_n and A_n be the partial transformation semigroup, (full) transformation semigroup, symmetric inverse semigroup, the symmetric group and the alternating group on X_n={1,…,n}. In this study we present the compilation of some studies which examine the algebraic structure of quasi-idempotents in P_n, T_(n,) 〖 I〗_n, S_n and A_n, and which obtaine the quasi-idempotent rank of these semigroups by using the (minimal) quasi-idempotent generating sets to the researchers.