WE-m semigroups (Q1061866)

If S is a semigroup, then let WE(S) denote the set of all positive integers m which satisfy the following condition: for every x,y\(\in S\), there is a positive integer k such that \((xy)^{m+k}=x^ my^ m(xy)^ k=(xy)^ kx^ my^ m\). It is proved that, for every semigroup S, WE(S) is a subsemigroup of the multiplicative semigroup of all positive integers. WE(S) is called the weak exponent semigroup of the semigroup S. A semigroup S is called a WE-m semigroup if \(m\in WE(S)\), supposing \(m\geq 2\). The main results of the paper are: 1. Every WE-m semigroup is a semilattice of archimedean WE-m semigroups. 2. A semigroup S is a WE-m archimedean semigroup with idempotent if and only if S is a retract extension of a completely simple E-m semigroup by a nil semigroup (A semigroup S is called an E-m semigroup if, for every x,y\(\in S\), \((xy)^ m=x^ my^ m\), supposing \(m\geq 2)\).