Homomorphisms into \(N^*\) (Q1849641)

The paper under review deals with continuous homomorphisms from \(\beta S\) into \(\mathbb N^*\) and from \(\mathbb N^*\) into \(\mathbb N^*\) and contains the Introduction and two sections. In Section 2 the authors study continuous homomorphisms from \(\beta S\) into \(\mathbb N^*\), where \(S\) is a discrete semigroup. They prove that if \(S\) is countable, then the smallest ideal \(K(\phi(\beta S))\) of \(\phi(\beta S)\), where \(\phi\colon \beta S\to\mathbb N^*\) is a continuous homomorphism, has the following properties: (i) every element of \(K(\phi(\beta S))\) is idempotent; (ii) \(K(\phi(\beta S))\) has only a finite number of minimal left ideals; (iii) every minimal right ideal in \(K(\phi(\beta S))\) is finite; (iv) \(K(\phi(\beta S))\) is compact; and (v) \(K(\phi(\beta S))\) is clopen in \(\phi(\beta S)\). Moreover, if \(S\) is countable and commutative, then (i) \(|K(\phi(\beta S))|=1\); (ii) the semiprincipal left ideals of the form \(\beta\mathbb N+n\), with \(x\in\phi(\beta S)\), are linearly ordered by inclusion; (iii) if \(q_1\) and \(q_2\) are idempotents in \(\phi(\beta S)\), then \(q_1=q_1+q_2\) or \(q_2=q_2+q_1\); (iv) if \(p_0\) is the unique minimal idempotent in \(\phi(\beta S)\), then \(p_0\) is algebraically a zero for \(\phi(\beta S)\) and topologically an isolated point in \(\phi(\beta S)\). In Section 3 the authors study continuous homomorphisms from \(\mathbb N^*\) into \(\mathbb N^*\) which do not arise from the conditions extensions of homomorphisms mapping \(\mathbb N\) to itself. They show that the image \(\phi(\mathbb N^*)\) under a homomorphism \(\phi\) of this kind has a unique idempotent \(e\) and \(\phi(\mathbb N^*)+\phi(\mathbb N^*)=\{ e\}\). Moreover, \(\phi(\mathbb N^*)\) has a unique smallest ideal and \(\mathbb N^*\) has no non-trivial retracts in the category of compact right topological semigroups.