I left this blog alone almost 2 years ago as I finished my “residency” at the algebra department. I was too busy with other things to even round out what I was doing then. In case you care, a paper came out about the rectangularity property.
But I have recently (end of 2012) started visiting the department regularly and have started work with Gerhard Wendt on some aspects of straucture theory in nearrings of automata. State Automata and Cellular Automata. While I would like to summarise it all for you now, I cannot and it is proving to be less difficult than I thought but more difficult than I hoped. Which is good I suppose.
But for a quick summary:
- Cellular or Tesselation automata over state groups with torsion free space groups are 2-primitive as nearrings.
- Over torsion space groups we get “really large” Jacobson radicals
- The Jacobson 2-radical of a state automata nearring lies within the set of “contraction” automata, i.e. those that never respond immediately to changes. We conjecture that it is precisely this and perhaps an old paper about topological nearrings from 1967 will help us work it out.
But I hope to post once a week (only one day of algebra per week, so perhaps it is still a theorem a day?) as I go on in order to try and summaruise what is going on. No pictures this week….
A Theorem a Day 