Crypto-preorders, topological relations, information and logic

ABSTRACT

As is well known, any preorder $R$ on a set $U$ induces an Alexandrov topology on $U$. In some interesting cases related to data mining an Alexandrov topology can be transformed into different types of logico-algebraic models. In some cases, (pre)topological operators provided by Pointless Topology may define a topological space on $U$ even if $R$ is not a preorder. If this is the case, then we call $R$ a crypto-preorder. The paper studies the conditions under which a relation $R$ is a crypto-preorder and how to transform a crypto-preorder into a proper preorder.

KEYWORDS:

• Alexandrov topology
• preorders
• pointless topology
• algebraic logic
• rough sets

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In modal terms, $〈A,B,R〉$ can be considered a sort of heterogeneous Kripke frame, with $[\cdot ]$, int and $\mathcal{C}$ in the role of necessity, and $〈\cdot 〉$, cl and $\mathcal{A}$ in the role of possibility.

2 Theorem 5.21 amends point (iv) of Corollary 1 of Pagliani (2014) and point (ii) of Facts 3 of Pagliani (2016), which state also the converse implication, erroneously.

3 Coprime filters are called prime filters in some texts.

4 $E_{\rho }$ is a generalisation of the Kolmogorov quotient and if ${\mathbf{L}}_{\mathcal{C}}$ is a topology, it is a result of the duality theorem about distributive lattices where any coprime (or prime) element x of the lattice is transformed into an ‘abstract point’ on which the elements of x collapse.

5 However the following is a direct proof. If $x{⪯}_{\left[i\right]}y$ then for all $X\subseteq U,x\in \left[i\right]\left(X\right)$ implies $y\in \left[i\right]\left(X\right)$. Therefore, $R^{⌣}\left(x\right)\subseteq X$ implies $R^{⌣}\left(y\right)\subseteq X$, all X. In particular, $R^{⌣}\left(x\right)\subseteq R^{⌣}\left(x\right)$ implies $R^{⌣}\left(y\right)\subseteq R^{⌣}\left(x\right)$. But the antecedent is true, so the consequence must be true, too. Since R is reflexive, so is $R^{⌣}$. Hence $y\in R^{⌣}\left(y\right)$ and, thus, $y\in R^{⌣}\left(x\right)$ so that $〈x,y〉\in R^{⌣}$. The opposite implication is proved analogously by transitivity: if $y\in R^{⌣}\left(x\right)$ and $R^{⌣}$ is transitive, then $R^{⌣}\left(y\right)\subseteq R^{⌣}\left(x\right)$. Hence, if $R^{⌣}\left(x\right)\subseteq X$ then $R^{⌣}\left(y\right)\subseteq X$, all $X\subseteq U$. Therefore, $x\in \left[i\right]\left(X\right)⟹y\in \left[i\right]\left(X\right)$. The thesis for $\left[e\right]$ and R is a trivial consequence.

6 This duality is a version of Birkhoff's representation theorem applied to finite topologies, that is, finite distributive lattices of sets. In fact from Corollary 5.20, for any $x\in U$, ${⪯}_{\omega }\left(x\right)$ is a co-prime filter (and ${⪯}_{\varkappa }\left(x\right)$ a prime ideal) and $E_{{\equiv }_{{⪯}_{\omega }}}\left(U\right)\cong \mathcal{J}\left({\mathrm{\Omega }}_{\tau }\right)$.

7 An even more general case, to be explored, is given when instead of a function a relation $Z\subseteq U\times U^{\prime }$ is used.

8 It is called a semi-topological formal system in Pagliani & Chakraborty, 2008

9 Since the monoid operation, ∩, is idempotent, a formal neighbourhood system is a quasi-topological formal system according to the taxonomy introduced in Pagliani and Chakraborty (2008).

10 Details may be found in Pagliani and Chakraborty (2008). Pay attention that in that book the apex R of ${\mathcal{N}}^R$ and the likes, is not used. A simplified proof can be found in Pagliani (2014).

11 In that book the proof of Proposition 15.14.9 suffers from some typos in the last line to be corrected as follows: ‘By (point-meet) $Y\cap Y^{\prime }\in \mathcal{Y}\cdot {\mathcal{Y}}^{\prime }$ and by N3, $Y\cap Y^{\prime }\in R\left(a\right)$, so that $X◂\mathcal{Y}\cdot {\mathcal{Y}}^{\prime }$ ’.

12 Enlarging $\mathrm{rng}\left(R1\right)$ does not work, because of the above constraints.

13 This is a direct proof of Proposition 15.14.14 of Pagliani and Chakraborty (2008) and is based on S-N2 instead of N2.