The near-ring of congruence-preserving functions on an expanded group.
Let \(\mathbf V\) be a finite expanded group, i.e., a group with possible further operations, e.g., a ring, or a group, or a near-ring, or an \(\Omega\)-group in the sense of \textit{A. G. Kurosh} [Lectures in general algebra (Oxford: Pergamon Press) (1965; Zbl 0105.24901)]. In the present paper, some information on the structure of the near-ring \(\langle C_0({\mathbf V});+,\circ\rangle\) of zero-preserving congruence-preserving functions on \(\mathbf V\) is obtained from investigating the ideal lattice \(\mathbf{Id\,V}\) of \(\mathbf V\). The main result (Theorem 5.1) is as follows. Suppose that \(A,B,C,D\) are ideals of \(\mathbf V\) such that \(A<B\), \(C<D\) with \(B/A\) and \(D/C\) being \(C_0(\mathbf V)\)-isomorphic and there are no ideals \(I,J\) of \(\mathbf V\) with \(A<I<B\), \(C<J<D\). Then the intervals \(I[A,B]\) and \(I[C,D]\) are projective intervals in \(\mathbf{Id\,V}\) in the sense of \textit{G. Grätzer} [General lattice theory, 2nd ed. (Basel: Birkhäuser) (1998; Zbl 0909.06002)]. The converse of this theorem also holds.
scientific article; zbMATH DE number 5004394
scientific article