Exploring Core Regular Double Stone Alegebras, CRDSA, I. (A note on CRDSA subalgebras and network node sets)
Daniel J. Clouse · arXiv (Cornell University) · 2018
This is the first in a series of three notes on an investigation into core regular double Stone algebras, CRDSA, and are meant to be read in order. They were born out of a model of network security where individual nodes are considered to be in one of 3 states. Applications of these results are being explored, results of at least one of which will be published at a later date. The two notes following this will establish a duality between the category of CRDSA and specifically crafted bi-topological spaces. Let J be any non-empty set of network nodes, not necessarily finite. We denote the node set bounded distributive lattice through the pairwise disjoint subsets of J with the well known binary operations of ternary set partitions and note J = 1 is our minimal case. We then show the resultant bounded distributive lattice is isomorphic to direct products of the 3 element chain, C_3. We then derive that every CRDSA is a subdirect product of C_3. We use these results along with a few known results to show the main result, namely Every Boolean algebra is the center of some core regular double Stone algebra, CRDSA. We then use that result to characterize all finite core regular double Stone Algebras, namely A CRDSA A is a subalgebra of C_3^J for some finite J if and only if it is isomorphic to C_3^K for some K less than or equal to J. Perhaps more importantly, we note that we have shown that any finite CRDSA is isomorphic to C_3^J for some finite J.