Some aspect of \(\theta\)-wreath sum of near-rings. (Q2844324)

The wreath sum of two near-rings \(N\) and \(K\) has been defined by \textit{M. N. Baruah} [``Near-rings and near-ring modules -- some special types'', PhD thesis, Gauhati University, Naya Prakash (1984)]. If \(K^N\) denotes the near-ring of all functions from \(N\) to \(K\) with respect to componentwise addition and multiplication, then \(K\text{\,Wr\,}N\) is the near-ring with additive group \((N,+)\oplus(K^N,+)\) and multiplication \((a,f)(b,g):=(ab,f^bg)\) where \(f^b\colon N\to K\) is the function defined by \(f^b(x)=f(bx)\) for all \(x\in N\) and \(f^bg\) is the multiplication in \(K^N\).NEWLINENEWLINE In this paper the authors define and study a generalization of \(K\text{\,Wr\,}N\). Let \(\theta\colon N\to N\) be an idempotent semigroup homomorphism (with respect to the multiplication on \(N\)). Then the \(\theta\)-\textit{wreath sum} of \(N\) and \(K\) has the same underlying group structure as \(K\text{\,Wr\,}N\), but the multiplication is given by \((a,f)\bullet^\theta(b,g):=(ab,f^b\bullet^\theta g)\) where \((f^b\bullet^\theta g)(x):=f(\theta(bx))g(x)\) for all \(x\in N\). The idea for this generalization is already present in an earlier paper of the first author but not referred to here [\textit{K. C. Chowdhury} and \textit{P. Das}, Southeast Asian Bull. Math. 36, No. 2, 169-185 (2012; Zbl 1265.16064)], where it was used in connection with near-ring groups (modules) on wreath sums. For a near-field \(K\), a restricted \(\theta\)-wreath sum of \(K\) and \(N\), as a subnear-ring of the \(\theta\)-wreath sum, is defined but contrary to what is claimed in the paper, this subset is neither closed under addition nor multiplication (and thus not a subnear-ring). A few results on substructures of and homomorphisms between wreath sums are given. The paper concludes with two lengthy examples, taken almost verbatim from the above mentioned paper. The purpose of repeating these examples here is not clear to this reviewer.