\(0\)-Archimedean semigroups (Q1917722)

Let \(S\) be a semigroup containing 0 i.e. \(S=S^0\). By a nil-ideal of a semigroup \(S\) we mean an ideal of \(S\) which is a nil-semigroup (that is, some power of every element is 0). Let \({\mathcal R}(S)\) be the union of all nil-ideals of \(S\) (i.e. the greatest nil-ideal of \(S\)). Define an equivalence relation \(\sigma_1\) on \(S\) by \(a\sigma_1b\) iff \(\Sigma_1(a)=\Sigma_1(b)\), \(a,b\in S\) where \(\Sigma_1(a)=\{x\in S:a\) divides some power of \(x\) denoted by \(a\to x\}\). A semigroup \(S=S^0\) is called 0-archimedean if \(a\to b\) for all \(a,b\in S-\{0\}\); weakly 0-archimedean if \(a\to b\) for all \(a,b\in S-{\mathcal R}(S)\). This paper characterizes the structure of weakly 0-archimedean semigroups and completely 0-archimedean semigroups. A completely 0-archimedean semigroup is a 0-archimedean semigroup containing a 0-primitive idempotent, where a 0-primitive idempotent is an idempotent which is minimal in the partially ordered set of all nonzero idempotents. As one of the theorems, \(S\) is weakly 0-archimedean if and only if \(S\) is an ideal extension of a nil-semigroup by a 0-archimedean semigroup, equivalently, \(S\) contains at most two \(\sigma_1\)-classes.