ALGSTR

Created by Ameyah Official

Overview of a two term lecture course in Algebra covering: Convolution Algebra and Algebraic Measurement Theory

#Algebra, #Free module, #Monoids, #Semiring-modules, #Semirings, #directed Graphs, #logistic networks

Vectorial map \vec

P set, p \in P, \vec : P^2 \to \Z^2

V1: \vec(\id(p)) = zero vector 0 \for all p \in P
V2: \vec((p,t)\ast\vec(t,q)) = \vec(p,t)+\vec(t,q)
for all p,t,q \in P

Basics

Set of maps btw sets: B^A

Algebraic Measurement Theory

Exm (\G_p,\M,\vec)

\G_p =

props funct. map

F1 \forall (a,b) \in E⁽²⁾: \Delta(a\ast b) = \Delta a \star \Delta b
F2 \forall p \in V : \Delta(\id p) = \epsilon

Maps

big and small kernels

Convolution Algebras

\Delta functorial map

\G = (G,\ast,\id) ANW with G = (V,E,\rho) graph\M = (M,\star,\epsilon)

\Delta: E \to M bzgl (\G,\M)

Measurement Setup \MM

\MM = (\G,\M,\Delta)
\G = Aktionsnetzwerk (Was messen)
\M = Monoid (Worin)
\Delta = functorial map (Wie)

linear maps

amicable pairs

Exm connection big and small kernel

Exm \SSS - \N = (\N,+,\cdot,0,1)

Monoid \M = (M,\star,\epsilon)

\M = (M,\star, \epsilon)
M = Menge
op: \star:M^2 \to M
\epsilon - ein Element \in M mit axioms
\forall a,b,c \in M
M1: (a \star b) \star c = a \star (b \star c)
M2: \epsilon \star a = a \star \epsilon

Exm ANW

Logistiker

ker f and \R^3/ker f

Semiring \SSS

additive commutative monoid
multiplicative commutative monoid

Aktionsnetzwerk \G = (G,\ast,\id) (ANW)

\G = (G,\ast,\id)
G = netzwerk = (V,E,\rho) aka directed graph
\ast weglassprod.
\id=identity

Weglassprod \ast

set partition A/\ker f

Exm

\N_add = (\N,+,0)
\Z_add = (\Z,+,0)
\Z_multi = (\Z,\cdot,1)
etc

Notirische Geschierzähler

Schreibtischtäter

commutative monoid \M = (M,\star,\epsilon))

forall a,b \in M: a \star b = b\star a

Polynome p: \N \to \S

\id = identity

directed graph G = (E,V,\rho)

G = (E,V,\rho)
V = vertices (e,f)
E= edges e,f\inE
\rho= map - E \to V^2
\rho forms two edges to one edge

logistic ANW

paths Path(G = (V,E,\rho)) --> path network

Path(G) = G^<+>:= (V,E^<+>,\rho^<+>)
- \rho<+>(e_1,...,e_n) = (\sigma e_1,...,\sigma e_n)
forall (e_1,...,e_n) \in E<+>