Euclidean nearrings with a proper nonzero closed connected right ideal and a left zero not in that ideal (Q1300188)

The Euclidean nearrings considered here are those nearrings whose additive group is isomorphic to \((\mathbb{R}^2,+)\). Such nearrings with a closed connected right ideal and a left zero not in that ideal are characterized. They are either left zero nearrings or have a multiplication defined in a simple way using continuous maps from \(\mathbb{R}\) to itself or to half lines. It is shown that without the condition on the left zero, the situation is much more complicated. The next stage is a determination of all possible isomorphisms between the nearrings in question. This leads on to determining the automorphism groups for a large family of these nearrings. All the right ideals of one family of nearrings are determined, then the same is done for the left ideals, hence all ideals are found. From this the structure of the quotient nearrings is determined. The left ideals and ideals in the remaining cases are described, as are the quotient nearrings. The structure of the multiplicative semigroups of these rings is studied next. Necessary and sufficient conditions for regularity are found, and Green's relations are determined. The maximal subgroups are described. The homeomorphisms and the automorphism groups of the multiplicative semigroups are determined. Putting all this together, it is shown that the multiplicative semigroups of the Euclidean nearrings under consideration are of a very restricted type: one of three possibilities. This is another substantial contribution to the study of Euclidean nearrings started by the author [in Monatsh. Math. 119, No. 4, 281-301 (1995; Zbl 0830.16032)].