Subject Classiication (ams)

A ?semigroup is a semigroup whose lattice of congruences is a chain with respect to inclusion. Schein 9] and Tamura 11] have investigated commutative ?semigroups, Trotter 13] exponential, Nagy 4] weakly exponential and Bonzini and Cherubini Spoletini 2] nite ?semigroups. The pupose of the present paper is to investigate archimedean weakly commutative and H?commutative ?semigroups. 1 Weakly commutative semigroups The theory of congruences on semigroups is fundamental and the study of semigroups, whose lattice of congruences is simple (in some sense), is therefore relevant. Important semigroups such as the full transformation semigroup, symmetric inverse semigroups and symmetric groups, on nite sets, are all ?semigroups (see 3], vol. II, page 247 and 12], examples 1 and 2). A semigroup S is weakly commutative if, for each a; b 2 S; there are x; y 2 S and an integer k > 0 such that (ab) k = xa = by ((6]). A semigroup S is H?commutative if, for each a; b 2 S, there are w; z 2 S 1 with ab = baw = zba 14]. Remark: A H?commutative semigroup is weakly commutative. Groups are H?commuta-tive semigroups and nilsemigroups are weakly commutative.