/pdf/on-the-sobrification-remainder-s-x-x-5aptuaorad.pdf

On the sobrification remainder $^{s}X-X$.

In this survey paper we give an historical and at the same time thematical overview of the development of ring geometry from its origin to the current state of the art. A comprehensive up-to-date list of literature is added with articles that treat ring geometry within the scope of incidence geometry.

On the history of ring geometry (with a thematical overview of literature)

Given a poset $P$, the set, $\Gamma(P)$, of all Scott closed sets ordered by
inclusion forms a complete lattice. A subcategory $\mathbf{C}$ of
$\mathbf{Pos}_d$ (the category of posets and Scott-continuous maps) is said to
be $\Gamma$-faithful if for any posets $P$ and $Q$ in $\mathbf{C}$, $\Gamma(P)
\cong \Gamma(Q)$ implies $P \cong Q$. It is known that the category of all
continuous dcpos and the category of bounded complete dcpos are
$\Gamma$-faithful, while $\mathbf{Pos}_d$ is not. Ho & Zhao (2009) asked
whether the category $\mathbf{DCPO}$ of dcpos is $\Gamma$-faithful. In this
paper, we answer this question in the negative by exhibiting a counterexample.
To achieve this, we introduce a new subcategory of dcpos which is
$\Gamma$-faithful. This subcategory subsumes all currently known
$\Gamma$-faithful subcategories. With this new concept in mind, we construct
the desired counterexample which relies heavily on Johnstone's famous dcpo
which is not sober in its Scott topology.

https://repository.nie.edu.sg/bitstream/10497/19501/1/LMCS-14-1_1.pdf

The Ho Zhao Problem

/pdf/construction-of-homomorphisms-of-m-continuous-lattices-1gt81epmto.pdf

Construction of homomorphisms of $M$-continuous lattices

An affine Hjelmslev plane is a near affine Hjelmslev plane with a parallelism. It is proved that every strongly n-uniform near affine Hjelmslev plane possesses an even parallelism and, if n > 2, uneven parallelisms as well. Secondly, we prove that a strongly n-uniform affine Hjelmslev plane A has an extension to a strongly n-uniform projective Hjelmslev plane if and only if the parallelism of A is even. The above results are applied to yield the first known examples of affine Hjelmslev planes which possess no extensions to projective Hjelmslev planes.

Existence of parallelisms and projective extensions for strongly n-uniform near affine Hjelmslev planes

https://www.ams.org/journals/tran/1965-120-01/S0002-9947-1965-0188963-7/S0002-9947-1965-0188963-7.pdf

On the representations of an abstract lattice as the family of closed sets of a topological space

Affine or projective Hjelmslev planes are called 1-uniform (also strongly 1-uniform) if they are finite customary affine or projective planes. If n > 1, an n-uniform affine or projective Hjelmslev plane is a (finite) Hjelmslev plane U with the following property: for each point P of ?, the substructure n1 p of all neighbor points of P is an (n 1)-uniform affine Hjelmslev plane. Associated with each point P is a sequence of neighborhoods lp C 2p C ... C np = 2X. For i k, ('i)j and (i k -).k are isomorphic. Then if 2I(i) denotes (i?)i_ 1(1), q... , I(n) is a periodic sequence of ordinary translation planes (all of order r) whose period is divisible by k. It is proved that if T 1 . 9 Tk is an arbitrary sequence of translation planes with common order and if n > k, then there exists a strongly n-uniform translation Hjelmslev plane U of width k such that I(i) T. for i < k. The proof of this result depends heavily upon a characterization of the class of strongly n-uniform translation Hjelmslev planes which is given in this paper. This characterization is given in terms of the constructibility of the n-uniform planes from the (n 1)-uniform planes by means of group congruences. Introduction. Hjelmslev planes are generalizations of customary affine and projective planes in which distinct lines may intersect in more than a single point. Throughout this paper, we will refer to a Hjelmslev plane as an H-plane. One calls a finite H-plane 1-uniform if it is a customary affine or projective H-plane: Received by the editors May 14, 1971. AMS (MOS) subject classifications (1970). Primary 05B25, 50D35; Secondary 05B30.

/pdf/the-structure-of-n-uniform-translation-hjelmslev-planes-2g5mkl0p0z.pdf

David A. Drake

Papers

On the representations of an abstract lattice as the family of closed sets of a topological space

The structure of $n$-uniform translation Hjelmslev planes