Chain/Hierarchy of Monoids - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:15:05Z http://mathoverflow.net/feeds/question/24723 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24723/chain-hierarchy-of-monoids Chain/Hierarchy of Monoids supercooldave 2010-05-15T10:45:40Z 2010-06-09T13:17:05Z <p>Let's assume that we have the following collection of structures:</p> <ul> <li>Some <em>space</em> $P$.</li> <li>Monoids $(M_{i+1},\circ_{i+1})$, and</li> <li>Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$</li> <li>And $\bullet_{0}:M_0\times P\to P$.</li> </ul> <p>satisfying</p> <ul> <li>($\bullet$ is a monoid action): $(m\circ_{i+1}m')\bullet_{i+1} n = m\bullet_{i+1}(m'\bullet_{i+1} n)$ and</li> <li>($m\bullet-$ is a homomorphism): $m\bullet_{i+1}(n\circ_{i}n')=(m\bullet_{i+1}n)\circ_{i} (m\bullet_{i+1} n')$.</li> </ul> <p>In my application, $P$ corresponds to computer programs. $M_0$ are modifications to elements of $P$. If you wish, you can think of $M_0$ as some kind of structured patch. Then each $M_{i+1}$ are <em>higher-order modifications</em> of the modifications in $M_i$.</p> <p>The hierarchy isn't necessarily infinite.</p> <p>I'm curious to know what kind of structure I'm looking at. I originally felt that I was defining some kind of $n$-category with one object at each level, namely the endomorphism, but one reader commented that my structures were too floppy, meaning that there were not enough equations.</p> <p>It seems that the structure I'm interested in is related to the <a href="http://www.math.rutgers.edu/~sthomas/book.ps" rel="nofollow">automorphism tower</a> for groups, except that I'm interest in monoids, and rather than automorphism, I'm only concerned with endomorphism, and I am working indirectly through monoid actions, rather than having the endomorphism apply to the morphisms at the level below.</p> <p>Have I defined a known structure? </p> <p>What natural equations would one expect to link the various levels with each other?</p> <p>What additional properties does it satisfy? What reasonable properties should it satisfy?</p> <p>Are there conditions under which it becomes degenerate?</p> <p>Any pointers would be appreciated.</p>