Ordered Desarguesian Affine Hjelmslev Planes

Abstract

The first chapter provides basic p~erequisites.In

Moulton Affine Hjelmslev Planes

“…We define a ternary relation p on the points of #f by (P 1 ,P 2 ,P 3 )ep (P t =(a i ,b i ), f = l,2, 3) if Pu ^2, ^3 I[™, n\ and b 1 <b 2 <b 3 or b t >b 2 > b 3 or if P l9 P 2 , P 3 I[m, n] 2 and a 1 <a 2 <a 3 or a x > a 2 > a 3 . It is easily verified that p satisfies the order axioms 01-07 given in [5]; in fact, p is equivalent to the ordering of X induced by the ordering of H in [5]. In particular, we have the following result.…”

“…An A.H. ring H with radical r\ is ordered if there exists a subset H + of H such that aeH implies exactly one of a e H + , a = 0, -a e H + holds; a,beH + implies a + beH + ; a,beH + and bé 17 imply ab e H + (cf. [5], 2.2). Let H~ = {aeH\-aeH + }.…”

“…planes are coordinatized by affine Hjelmslev rings (A.H. rings) which are local rings whose radicals are equal to their sets of two-sided zero divisors and whose principal right ideals are totally ordered. In [5], ordered D.A.H. planes were defined and the induced orderings of their A.H. rings were discussed.…”

“…The relation p' clearly satisfies axioms 01-06 (cf. [5]), so it remains to verify 07. Since p is an ordering of Sff, any non-degenerate parallel projection which does not involve lines of the third kind will preserve the ordering p'.…”

Ordered Line and Skew-Fields in the Desargues Affine Plane

“…Considerable work on ordered plane geometries has been done (see, e.g., J. Lipman [9], V.H. Keiser [6], H. Tecklenburg [17] and L. A. Thomas [18,19]).…”