On near-ring idempotents and polynomials on direct products of \(\Omega\)-groups (Q2747928)

Let \(N\) be a right zero-symmetric near-ring with identity. A unital \(N\)-group \(\Gamma\) is called tame [\textit{S. D. Scott}, Proc. Edinb. Math. Soc., II. Ser. 23, 275-296 (1980; Zbl 0461.16027)] if and only if for all \(\gamma,x\in\Gamma\), \(n\in N\) there is an element \(m\in N\) such that \(n(\gamma+x)-n\gamma=mx\). Note that in a tame \(N\)-group, all \(N\)-subgroups are ideals. In the lattice of ideals of \(\Gamma\), \(A\) is a subcover of \(B\) if \(A\subset B\) and there is no ideal strictly between them. Write \(A=B^-\) if \(B\) is join irreducible and \(A\) is the unique subcover of \(B\). The first major result in this paper is the following theorem. Let \(N\) be a zero-symmetric near-ring with identity, and let \(\Gamma\) be a tame \(N\)-group. Assume that \(N\) has the DCC on left ideals and that the ideal lattice of \(\Gamma\) satisfies both the ACC and the DCC. Let \(H\) be an \(N\)-subgroup of \(\Gamma\). Then the following are equivalent. (1) There is an element \(e\in N\) with \(eh=h\) for all \(h\in H\) and \(e\Gamma\subseteq H\). (2) If \(A\) and \(D\) are join irreducible ideals of \(\Gamma\) and if \(A\subseteq H\) and the \(N\)-groups \(A/A^-\) and \(D/D^-\) are \(N\)-isomorphic, then \(D\subseteq H\). The proof is quite deep and requires a careful analysis of the ideal structure of \(\Gamma\). Several interesting results are proved along the way.NEWLINENEWLINENEWLINEThe second part of the paper (and the title) concerns polynomial functions on an \(\Omega\)-group \(V\). The main result of the first section is applied to the near-ring \(P_0(V)\) of zero-preserving functions on \(V\). If \(V\) is a group without further operations then \(P_0(V)=I(V)\), the inner automorphism near-ring. Using \textit{S. D. Scott} [Monatsh. Math. 73, 250-267 (1969; Zbl 0182.35201)], it can be shown that \(P_0(G_1\times G_2)\cong P_0(G_1)\times P_0(G_2)\) if and only if \(G_1\times G_2\) has no skew congruences, which is the case if and only if every normal subgroup of \(G_1\times G_2\) is equal to \(N_1\times N_2\) for some normal subgroups \(N_1\) of \(G_1\) and \(N_2\) of \(G_2\). Here, the author shows that if \(V=\prod^k_{i=1}V_i\), where the \(\Omega\)-groups \(V_i\) are similar, then all the projection functions are in \(P_0(V)\) if and only if every ideal of \(V\) is the direct product of ideals in each \(V_i\). It is then an easy consequence of this that if \(V\) has no skew-congruences then there is a bijective mapping \(\text{Pol}_n(\prod^k_{j=1}V_j)\to\prod^k_{j=1}\text{Pol}_nV_j\) where \(\text{Pol}_nV\) is the set of all \(n\)-ary polynomial functions from \(V^n\) to \(V\).