MathOverflow Asked by Tegiri Nenashi on January 1, 2022
Starting with fundamental group, say of a circle, let’s reflect back to path groupoid a little. The path concatenation operation is partial, but this can be remedied by focusing on the sets of paths, rather than individual paths. Then, if the two sets of paths of the multiplier have no end points coinciding with start points of the multiplicand, then the result is the empty set, which is the additive identity of the ring. Therefore, Boolean algebra of sets provides the additive structure of the ring.
The ring multiplication is inherited from groupoid, that is path concatenation, and it is non commutative. There is a multiplicative identity, which is the set of trivial loops around all points in the space.
To prove the distributivity property, let $P$,$Q$,$R$ be sets of paths. Then
$P (Q cup R) = {p cdot x | p in P wedge x in Q cup R } = $
$= {p cdot x | (p in P wedge x in Q) vee (p in P wedge x in R) } = $
$= {p cdot x | (p in P wedge x in Q) } cup {p cdot x | (p in P wedge x in R) } $
At this point, it is not immediately obvious if any congruence is needed to be introduced. Yet, I’m puzzled why there appears nothing on the web matching the "fundamental ring" search term. Or there is? And what is the ring of path sets of the circle, is it a familiar mathematical object?
1 Asked on January 9, 2021 by ben-macadam
0 Asked on January 8, 2021 by user158773
complex geometry complex manifolds cv complex variables riemann surfaces vector bundles
0 Asked on January 8, 2021 by ma-joad
fa functional analysis functional calculus sp spectral theory unbounded operators
1 Asked on January 7, 2021 by b-merlot
0 Asked on January 7, 2021
filtrations markov chains measure theory pr probability stochastic processes
2 Asked on January 6, 2021 by logictheorist
1 Asked on January 5, 2021 by thiku
1 Asked on January 5, 2021 by cabbage
0 Asked on January 4, 2021 by qiaochu-yuan
1 Asked on January 4, 2021 by adam-p-goucher
0 Asked on January 4, 2021 by seddik-merdaci
1 Asked on January 4, 2021 by vassilis-papanicolaou
1 Asked on January 3, 2021 by user97621
1 Asked on January 2, 2021 by user666
finite groups modules noncommutative algebra ra rings and algebras rt representation theory
47 Asked on January 2, 2021 by roman-starkov
0 Asked on January 1, 2021 by user263322
algorithms convex optimization limits and convergence nonlinear optimization
1 Asked on December 31, 2020 by bernied
4 Asked on December 30, 2020 by gottfried-helms
2 Asked on December 30, 2020 by user163840
Get help from others!
Recent Questions
Recent Answers
© 2023 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP