Kernels of fibered groups with operators
Let \(<G,F>\) be a fibered group, \(F=\{G_ i\}\), and let S be a monoid (with zero) of operators on \(<G,F>\). The kernel of the fibered group \(<G,F>\) with operators S is the near-ring \(M_ S(G,F)=\{f: G\to G|\) \(\forall G_ i\in F\), \(f(G_ i)\subseteq G_ i\) and \(\forall \sigma \in S\), \(f\sigma =\sigma f\}\). Throughout the paper all structures are assumed to be finite. The paper is concerned with determining when \(M_ S(G,F)\) is a near-field, local near-ring or simple near-ring for the special choices (i) S is a group (with zero) of automorphisms and (ii) S is a cyclic semigroup with zero and identity. Several characterization theorems are obtained of which the following are a sample. Theorem. When \(S=A\) is a group (with zero) of automorphisms then \(M_ A(G,F)\) is a local near-ring if and only if \(M_ A(G,F)\) is a near- field. The second possibility, \(S=<\alpha >\cup \{0,id\}\), divides into two cases. Theorem. If \(\alpha\) is neither invertible nor nilpotent then \(M_ S(G,F)\) is neither local nor simple. When \(\alpha\) is nilpotent, an equivalence relation is defined on G whose classes are called components. If G has just one component then G is connected. A subset \(Y\subset G^*\), \(Y=\{y_ 1,y_ 2,...,y_ k\}\) is a generating set if \(G=\{0\}\cup Sy_ 1\cup...\cup Sy_ k\) and \(Sy_ i\varsubsetneq \cup_{j\neq i}Sy_ j\). Theorem. If \(<G,F>\) is connected then (a) ker \(\alpha\) \(\subseteq G_ i\) for some i, say \(i=0\). (b) If Y is an arbitrary but fixed generating set then the following are equivalent (1) \(Y\cap G_ 0=\emptyset\); (2) \(M_ S(G,F)\) is a local near-ring; (3) \(M_ S(G,F)\) is a near-field; (4) \(M_ S(G,F)\) is a simple near-ring; (5) \(M_ S(G,F)\) is a 2-semisimple near-ring; (6) \(M_ S(G,F)\cong {\mathbb{Z}}_ 2\). In the non-connected case, a characterization of \(M_ S(G,F)\) in terms of the fibration F is obtained for \(M_ S(G,F)\) to be simple.
scientific article; zbMATH DE number 3964112
scientific article