Homomorphisms of nearrings of continuous functions from topological spaces into the asymmetric nearring (Q1304879)

A topological nearring is given as a triple \((N,+,\cdot)\), where \((N,+)\) is a topological group, \((N,\cdot)\) is a topological semigroup, and one distributive law is valid: \((a+b)\cdot c=a\cdot c+b\cdot c\). This generalizes the notion of topological nearfield. In contrast to the situation with nearfields, the additive group of a nearring may well be nonabelian. It may also happen that there exists \(a\in N\) with \(a\cdot 0\neq 0\) (for continuity reasons, the usual precaution of defining these products to be zero is not sensible). Nearrings with \(N\cdot 0=\{0\}\) are called zero symmetric. Up to isomorphism, there is exactly one topological nearring \({\mathcal N}\) with identity which is not zero symmetric and whose additive group is the one of \({\mathbb R}^2\): the multiplication is given by \((v_1,v_2)\cdot(w_1,w_2)=(v_1w_1,v_1w_2+v_2)\) [the author, Acta Sci. Math. 62, No. 1-2, 115-125 (1996; Zbl 0862.16029)]. In analogy to the theory for the ring \({\mathcal C}(X)\) of all continuous functions from a topological space \(X\) to the real field \({\mathbb R}\), the author investigates the nearring \({\mathcal N}(X)\) of all continuous functions from \(X\) to \({\mathcal N}\). For homeomorphic spaces \(X,Y\) the isomorphisms from \({\mathcal N}(X)\) onto \({\mathcal N}(Y)\) are completely described (Thm. 2.2). More generally, homomorphisms between such nearrings of functions are described in Thm. 2.3: these are induced in the usual (contravariant) manner by continuous maps defined on closed-open subsets; however, additional distortion by a continuous function into the reals may take place. In the third section of the paper under review, the author determines the semigroup \(\text{End}({\mathcal N}(X))\) of endomorphisms of \({\mathcal N}(X)\), the subsemigroup \(\text{End}_{FI}({\mathcal N}(X))\) consisting of those endomorphisms that fix the identity of \({\mathcal N}(X)\), and the group \(\Aut({\mathcal N}(X))\). It is shown (Thm. 3.11, 3.13) that, for the class of compact spaces as well as for the class of realcompact equalizer spaces in the sense of [\textit{J. C. Warndof}, Fundam. Math. 66, 25-43 (1969; Zbl 0189.23201)], the space \(X\) is determined by the isomorphism type of the semigroup \(\text{End}({\mathcal N}(X))\), as well as by the isomorphism type of the semigroup \(\text{End}_{FI}({\mathcal N}(X))\).