Quora: What is an ALGEBRA structure

What is an algebra? by Tikhon Jelvis https://www.quora.com/What-is-an-algebra/answer/Tikhon-Jelvis?ch=3&share=2dd8711d&srid=oZzP

“Basically, an algebra is just an algebraic structure. It’s some set A along with some number of functions closed over the set. It’s a generalization over the structures we normally study: a group is an algebra, a ring is an algebra, a lattice is an algebra… etc.

Algebras have different “signatures” which specify the functions it has. For example, a group is an algebra that has an identity element, a function of one argument and a function of two arguments.

That is, a group with a carrier set A is just a tuple:
⟨A, 0:A, −:A→A, +:A×A→A⟩

For uniformity, we can write all of these as functions in the form An→A, where n is the “arity” of a function—the number of arguments it has. The identity element is a function A0→A, which just identifies a single element from A. Thus, we can talk about the signature of an algebra as the arities of its functions.

A group would be (0, 1, 2) while a ring would be (0, 0, 1, 2, 2).

Generally, the functions of an algebra have to be associative. Sometimes, we also look at other laws—for example, we might want to study algebras with commutative operations like Abelian groups.

So the intuition for an algebra in general is that it’s any structure like a group, a ring or whatever else we like. As the name “structure” implies, these additional operations on a set expose the internal structure of its elements: a group describes symmetries, a lattice describes a partial order and so on.

The study of algebras, then, can be thought of as the study of “structured sets” in general.”

BM Category Theory II 8: F-Algebra, Lambek’s Lemma , Catamorphism, Coalgebra, Anamorphism

[Continued from previous BM Category Theory …]

\boxed { \text {type Algebra f a = f a} \to \text {a} }

Intuition: [Artificial Intelligence] You teach the computer like to a Primary 6 kid, that Algebra is a type of expression (f) which, after evaluation,  returns a value.

If a = i (initial) [or u (terminal)],
\boxed { \text {(f i} \to \text {i )} \implies \text {f = Fix-point} }

Intuition: Fix-point because, the Initial “i”, after evaluating the expression f, returns itself “i”.

Lambek’s Lemma 
\boxed { \text {Initial Algebra is an Isomorphism} }

Note: Endo-functor is a functor (equivalent to function in Set Theory) within the same Category (Endo = Self = 自)

Video 8.1 F-Algebras & Lambek’s Lemma 

Video 8.2 Catamorphism & Anamorphism 

foldr ~ catamorphism (浅层变质) of a Fix-point endo-functor on a List.

Examples: Fibonacci, Sum_List

Remark: Cool Math! the more  advanced concept it is, the more closer to Nature (eg.Geology, Biology) : Catamorphism 浅层(风化)变质, or “thin-layer change in nature” (in Functional Programming languages: foldr or map) eg : add1 to a list (1 5 3 8…) 
= (2 6 4 9 …)

\boxed { \text {type Coalgebra f a = a} \to \text {f a} }

Intuition: Reverse of Algebra, given a value, Coalgebra returns an expression (f).

Anamorphism (合成变质) ~ unfoldr

Example: Prime numbers

Remark: Anamorphism (合成变质) or “synthesised change in nature“: eg. Start from a  “seed” prime number “2” generates  all other infinite prime numbers (3 5 7 9 11 13 17 …)

Note: In Haskell, no difference between Initial and Terminal Fix-points. However, since Fix-point is not unique, in Category Theory there is the Least Fix-point (Initial) and Greatest Fix-point (Terminal).


Reading “Understanding F-Algebra ” by BM: https://bartoszmilewski.com/2013/06/10/understanding-f-algebras/

Catamorphism (下态) : https://www.zhihu.com/question/52180621/answer/129582557

Anamorphism : https://zhuanlan.zhihu.com/cofree/21354189

F-Algebra & F-coalgebra: http://stackoverflow.com/a/16022059/5822795

Initiation aux Mathématiques Modernes – (TV 1969)

This French Classic (8 series lectures) on “Introduction to Modern Maths” made in 1969 is still valid today for Modern Maths students who need to be “initialized” for abstract, rigorous math, different from high-school maths.

Partie 1 : ensembles et parties

View the other 7 series below:

Initiation à l’algèbre (TV 1969): http://www.youtube.com/playlist?list=PLGwb7STcll74lv38PsnYmB4sr5fpWTEsq

Analysis -> (Topology) -> Algebra

Mathematics is divided into 2 major branches:
1. Analysis (Continuity, Calculus)
2. Algebra (Set, Discrete numbers, Structure)

In between the two branches, Poincaré invented in 1900s the Topology (拓扑学) – which studies the ‘holes’ (disconnected) in-between, or ‘neighborhood’.

Topology specialised in
–  ‘local knowledge’ = Point-Set Topology.
– ”global knowledge’ = Algebraic Topology.

The local data of consumer behavior uses ‘Point-Set Topology’; the global one is ‘BIG Data’ using Algebraic Topology.

Post-Modern Algebra

Modern Algebra: Based on the 1931 influential book “Modern Algebra” written by Van de Waerden (the student of E. Noether). Pioneered by the 20th century german Göttingen school of mathematicians, it deals with Mathematics in an abstract, axiomatic approach of mathematical structures such as Group, Ring, Vector Space, Module and Linear Algebra. It differs from the computational Algebra in 19th century dealing with Matrices and Polynomial equations.

This phase of Modern Algebra emphasises on the algebraization of Number Theory: {N, Z, Q, R, C}

Post-Modern Algebra: The axiomatic, abstract treatment of Algebra is viewed as boring and difficult. There is a renewed interest in explicite computation, reviving the 19th century invariant theory. Also the structural coverage (Group, Ring, Fields, etc) in Modern Algebra is too narrow. There is emphasis on other structures beyond Number Theory, such as Ordered Set, Monoid, Quasigroup, Category, etc.

The non-abelian Group S3 (order 6) has structures appeared in many branches of maths and sciences:

[Reference] :
Post-Modern Algebra” by
Jonathan D.H. Smith and Anna B. Romanowska
1999, John Wiley & Sons

Math Duality

Mathematics is roughly divided into 2 categories:

‘Macro’ Math: Algebra

‘Micro’ Math: Analysis (or the outdated name Calculus)

Algebra has been transformed rapidly from 19th century after Galois’s invention of Group Theory, and expanded by David Hilbert and his students E. Noether, Artin, etc in Axiomatic Algebra, takes a very macro view of Mathematical structures in abstract thinking.

Analysis, also after 19th century Cauchy and Wierestrass’s invention of ‘epsilon-delta’ micro view of Calculus, transformed the Newton Calculus into rigourous Math.

The old school of division of Pure and Applied Math is no longer valid. Take for example, the Applied Math used in Google Search Algorithm uses abstract Vector Space of Matrices in Linear Algebra (Pure Math).

Chinese School Algebra

A UK boy studying Grade 7 Algebra in a Chinese school: a culture shock, not only linguistic but pedagogical:


Chinese Math emphasizes on drilling and computational techniques, which earns them 2 decades of championship in the International Math Olympiad
It has its weakness, though, in abstract mathematical thinking and theoretical foundation – as found strongly in the French Math – which limits China from producing any Fields Medalist todate. (2 Overseas Chinese Fields medalists are ST Yau 丘成桐 and Terence Tao, 陶哲轩 who were both educated outside of China: Yau in HK/USA and Tao in Australia/USA)