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SandboxDemoFreeMonoid/-Module

Edit detail for SandboxDemoFreeMonoid/-Module revision 1 of 1

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Editor: pagani Time: 2020/01/30 19:42:09 GMT+0
Note:

changed:
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\begin{axiom}
-- FreeMonoid(S: SetCategory)
-- S can be almost anything

X:=FreeMonoid(Symbol)
e:=[e[i]::X for i in 1..5]

-- Some terms
t1:= e.1*e.2
t2:= t1^3

-- Equality
eq1:=t1=e.2*e.1
test(eq1)

eq2:=t2=t1*t1*t1
test(eq2)

-- Any symbol admissible ;)
s::X
s*t1 
a*%
first %

-- Methods
varList t1
varList t2

retractIfCan t1
retract e.1

length t1
length t2
size t2

factors t2

-- etc.

-- Restrict to b_1, b_2, b_3.
B:=OrderedVariableList [b[i] for i in 1..3]

Y:=FreeMonoid(B)
b:=[x::Y for x in enumerate()$B]
t3:=b.1*b.2*b.3*b.1
factors t3
t3*b.1  -- note exps

test(b.1*b.2=b.2*b.1)

--  FreeModule(R: Join(SemiRng,AbelianMonoid),S: SetCategory) 
--  combine

R:=Expression Integer
Z:=FreeModule(R,Y)

bb:=[x::Z for x in b]

t4:=bb.1 + t3::Z
t5:=sin(x*y)*t4+z^k*bb.2

-- Conclusion: combine FreeMonoid/FreeModule -> Algebra impl.
-- Products e.g 'product' from GradedAlgebra then take quotients.


\end{axiom}

fricas -- FreeMonoid(S: SetCategory)

S can be almost anything
X:=FreeMonoid(Symbol)

$\label{eq1}\hbox{\axiomType{FreeMonoid}\ } (\hbox{\axiomType{Symbol}\ })$

(1)

Type: Type

fricas

e:=[e[i]::X for i in 1..5]

$\label{eq2}\left[{e_{1}}, \:{e_{2}}, \:{e_{3}}, \:{e_{4}}, \:{e_{5}}\right]$

(2)

Type: List(FreeMonoid(Symbol))

fricas

-- Some terms
t1:= e.1*e.2

$\label{eq3}{e_{1}}\ {e_{2}}$

(3)

Type: FreeMonoid(Symbol)

fricas

t2:= t1^3

$\label{eq4}{e_{1}}\ {e_{2}}\ {e_{1}}\ {e_{2}}\ {e_{1}}\ {e_{2}}$

(4)

Type: FreeMonoid(Symbol)

fricas

-- Equality
eq1:=t1=e.2*e.1

$\label{eq5}{{e_{1}}\ {e_{2}}}={{e_{2}}\ {e_{1}}}$

(5)

Type: Equation(FreeMonoid(Symbol))

fricas

test(eq1)

$\label{eq6} \mbox{\rm false}$

(6)

Type: Boolean

fricas

eq2:=t2=t1*t1*t1

$\label{eq7}{{e_{1}}\ {e_{2}}\ {e_{1}}\ {e_{2}}\ {e_{1}}\ {e_{2}}}={{e_{1}}\ {e_{2}}\ {e_{1}}\ {e_{2}}\ {e_{1}}\ {e_{2}}}$

(7)

Type: Equation(FreeMonoid(Symbol))

fricas

test(eq2)

$\label{eq8} \mbox{\rm true}$

(8)

Type: Boolean

fricas

-- Any symbol admissible ;)
s::X

$\label{eq9}s$

(9)

Type: FreeMonoid(Symbol)

fricas

s*t1

$\label{eq10}s \ {e_{1}}\ {e_{2}}$

(10)

Type: FreeMonoid(Symbol)

fricas

a*%

$\label{eq11}a \ s \ {e_{1}}\ {e_{2}}$

(11)

Type: FreeMonoid(Symbol)

fricas

first %

$\label{eq12}a$

(12)

Type: Symbol

fricas

-- Methods
varList t1

$\label{eq13}\left[{e_{1}}, \:{e_{2}}\right]$

(13)

Type: List(Symbol)

fricas

varList t2

$\label{eq14}\left[{e_{1}}, \:{e_{2}}\right]$

(14)

Type: List(Symbol)

fricas

retractIfCan t1

$\label{eq15}\verb#"failed"#$

(15)

Type: Union("failed",...)

fricas

retract e.1

$\label{eq16}e_{1}$

(16)

Type: Symbol

fricas

length t1

$\label{eq17}2$

(17)

Type: PositiveInteger?

fricas

length t2

$\label{eq18}6$

(18)

Type: PositiveInteger?

fricas

size t2

$\label{eq19}6$

(19)

Type: PositiveInteger?

fricas

factors t2

$\label{eq20}\begin{array}{@{}l} \displaystyle \left[{\left[{gen ={e_{1}}}, \:{exp = 1}\right]}, \:{\left[{gen ={e_{2}}}, \:{exp = 1}\right]}, \:{\left[{gen ={e_{1}}}, \:{exp = 1}\right]}, \right. \ \ \displaystyle \left.\:{\left[{gen ={e_{2}}}, \:{exp = 1}\right]}, \:{\left[{gen ={e_{1}}}, \:{exp = 1}\right]}, \:{\left[{gen ={e_{2}}}, \:{exp = 1}\right]}\right]$

(20)

Type: List(Record(gen: Symbol,exp: NonNegativeInteger?))

fricas

-- etc.

-- Restrict to b_1, b_2, b_3.
B:=OrderedVariableList [b[i] for i in 1..3]

$\label{eq21}\hbox{\axiomType{OrderedVariableList}\ } ([ b [ 1 ] , b [ 2 ] , b [ 3 ] ])$

(21)

Type: Type

fricas

Y:=FreeMonoid(B)

$\label{eq22}\hbox{\axiomType{FreeMonoid}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ b [ 1 ] , b [ 2 ] , b [ 3 ] ]))$

(22)

Type: Type

fricas

b:=[x::Y for x in enumerate()$B]

$\label{eq23}\left[{b_{1}}, \:{b_{2}}, \:{b_{3}}\right]$

(23)

Type: List(FreeMonoid(OrderedVariableList([b[1],b[2],b[3]])))

fricas

t3:=b.1*b.2*b.3*b.1

$\label{eq24}{b_{1}}\ {b_{2}}\ {b_{3}}\ {b_{1}}$

(24)

Type: FreeMonoid(OrderedVariableList([b[1],b[2],b[3]]))

fricas

factors t3

$\label{eq25}\begin{array}{@{}l} \displaystyle \left[{\left[{gen ={b_{1}}}, \:{exp = 1}\right]}, \:{\left[{gen ={b_{2}}}, \:{exp = 1}\right]}, \:{\left[{gen ={b_{3}}}, \:{exp = 1}\right]}, \right. \ \ \displaystyle \left.\:{\left[{gen ={b_{1}}}, \:{exp = 1}\right]}\right]$

(25)

Type: List(Record(gen: OrderedVariableList([b[1],b[2],b[3]]),exp: NonNegativeInteger?))

fricas

t3*b.1  -- note exps

$\label{eq26}{b_{1}}\ {b_{2}}\ {b_{3}}\ {{b_{1}}^{2}}$

(26)

Type: FreeMonoid(OrderedVariableList([b[1],b[2],b[3]]))

fricas

test(b.1*b.2=b.2*b.1)

$\label{eq27} \mbox{\rm false}$

(27)

Type: Boolean

fricas

--  FreeModule(R: Join(SemiRng,AbelianMonoid),S: SetCategory) 
--  combine

R:=Expression Integer

$\label{eq28}\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ })$

(28)

Type: Type

fricas

Z:=FreeModule(R,Y)

$\label{eq29}\hbox{\axiomType{FreeModule}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{FreeMonoid}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ b [ 1 ] , b [ 2 ] , b [ 3 ] ])))$

(29)

Type: Type

fricas

bb:=[x::Z for x in b]

$\label{eq30}\left[{b_{1}}, \:{b_{2}}, \:{b_{3}}\right]$

(30)

Type: List(FreeModule(Expression(Integer),FreeMonoid(OrderedVariableList([b[1],b[2],b[3]]))))

fricas

t4:=bb.1 + t3::Z

$\label{eq31}{{b_{1}}\ {b_{2}}\ {b_{3}}\ {b_{1}}}+{b_{1}}$

(31)

Type: FreeModule(Expression(Integer),FreeMonoid(OrderedVariableList([b[1],b[2],b[3]])))

fricas

t5:=sin(x*y)*t4+z^k*bb.2

$\label{eq32}{{\sin \left({x \ y}\right)}\ {b_{1}}\ {b_{2}}\ {b_{3}}\ {b_{1}}}+{{\sin \left({x \ y}\right)}\ {b_{1}}}+{{{z}^{k}}\ {b_{2}}}$

(32)

Type: FreeModule(Expression(Integer),FreeMonoid(OrderedVariableList([b[1],b[2],b[3]])))