🟥 🟥 No code tool for Abstract Algebra

[daijapan]

current solution : M2, GAP, SageMath https://www.unimelb-macaulay2.cloud.edu.au/#home this is how it works but hard to use https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/PolyominoIdeals/html/_polyo__Matrix.html

(below is Rizwan's past work - we can make this better)

PolyominoIdeals: a package to deal with polyomino ideals, Macaulay2, available at https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/PolyominoIdeals/html/index.html

[daijapan]

A - how to count - 1-2 B- how to count - 1-2-3 C-

[daijapan]

D - example of small ones E - middle size one F - prime vs non-prime - ideal

[daijapan]

G - M2 example, tree shape , input, process , output H - graph theory vs Abstract algebra - cross section I - M2 vs GAP vs SageMath - comparison - what are PQRS?

[daijapan]

K - ideal is prime or non-prime? L - no prime with hole, prime without hole

[RizwanJdr]

as a starting point we can start with graphs and monomial ideas associated to graphs because the binomials ideals are complex cases. Once we have functional environment for monomial ideals we will try to extend that functionality to binomials

The most common monomial ideal associated to a graph is called edge ideal and defined as below:

Let G be a graph and we have edges of G labeled by e1, e2, e3,..., en now suppose that e1 is the edge between vertex v1 and v1' similarlry e2 is edge between v2 and v2'

we wil define a monomial associated to each edge as x1x1' associated to e1 x2x2' associated to e2 and so on

the ideal generated by all these monomials is called edge ideal

[daijapan]

Great - thank you

Dave Ishii, Mathematician 🔢 石井 大輔 数学者

Founder & CEO, Kiara Inc.

We make the better UX in workplace & finance with AI

On Fri, Feb 9, 2024 at 8:57 Rizwan @.***> wrote:

as a starting point we can start with graphs and monomial ideas associated to graphs because the binomials ideals are complex cases. Once we have functional environment for monomial ideals we will try to extend that functionality to binomials

The most common monomial ideal associated to a graph is called edge ideal and defined as below:

Let G be a graph and we have edges of G labeled by e1, e2, e3,..., en now suppose that e1 is the edge between vertex v1 and v1' similarlry e2 is edge between v2 and v2'

we wil define a monomial associated to each edge as x1x1' associated to e1 x2x2' associated to e2 and so on

the ideal generated by all these monomials is called edge ideal

— Reply to this email directly, view it on GitHub https://github.com/daijapan/Kiara-University/issues/15#issuecomment-1935114468, or unsubscribe https://github.com/notifications/unsubscribe-auth/ACQNVFOSB3EEKUDJCY3Q2STYSVQ7ZAVCNFSM6AAAAABB73AP62VHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMYTSMZVGEYTINBWHA . You are receiving this because you authored the thread.Message ID: @.***>

[daijapan]

@RizwanJdr update

[daijapan]

note : this should be next -gen ver of this (Rizwan's work) PolyominoIdeals: a package to deal with polyomino ideals, Macaulay2, available at https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/PolyominoIdeals/html/index.html

[daijapan]

@RizwanJdr hey let's talk today

[daijapan]

@RizwanJdr
A - pitch very specific mini size example (2-3 patterns) B- identify what are vague - try to specify C- put difficult ones in icebox / pend box D - start ASAP

[RizwanJdr]

a basic concept of this idea

[RizwanJdr]

an important thing is here that the missing cells are not necessarily to be inside to make a topological hole. They could be on the corners as well.

We are interested to read homological and algebraic properties of the ideals generated by the associated set of binomials through the shape of these polyominoes.

For example one major concept that is proven is that if the missing cells are not in the center that is there is no hole in the polyomino then the ideal associated is prime ideal.

An other interesting concept is that if we arrange maximum number of rooks (as of chess) on this polyomino in a way that no two rooks are attacking each other then the regularity (an algebraic concept) is equal to the these number of rooks.

[RizwanJdr]

Existing algo is time consuming and not efficient because there are still several manual steps.

[RizwanJdr]

Existing soltion: Sketch the polyomino on geogebra

Get the labels from geogebra or manually create a list

Use this list as input in M2

[daijapan]

@RizwanJdr tks - can you copy and paste javascript from geogebra and code from M2? so that we can customize?

[RizwanJdr]

the ideal associated to polyomino on the left is prime but it is not prime for the polyomino on the right

[RizwanJdr]

I am using geogebra to sketch these (https://www.geogebra.org/download?lang=en)

[RizwanJdr]

The generators of ideal of the polyomino on the left side are

[RizwanJdr]

The generators of the ideal of the polyomino on the right side are:

[RizwanJdr]

the complete code implemented in M2 is :

`loadPackage "PolyominoIdeals"; Q={{{1,1},{2,2}},{{2,1},{3,2}},{{3,1},{4,2}},{{1,2},{2,3}},{{3,2},{4,3}},{{1,3},{2,4}},{{2,3},{3,4}},{{3,3},{4,4}}}; I = polyoIdeal Q; g1 = gens I

Q2 = {{{2,1},{3,2}},{{3,1},{4,2}},{{4,2},{5,3}},{{4,3},{5,4}},{{3,4},{4,5}},{{2,4},{3,5}},{{1,3},{2,4}},{{1,2},{2,3}}}; I2 = polyoIdea Q2; g2 = gens I2`