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## Main Question or Discussion Point

I just found out that the words "algebraic structure" have a precise definition and that this

notion is not just common language!

I am thinking that it follows from the idea of a "structured set".

Unfortunately, I can find basically nothing on this concept from browsing online.

The only sources I have found are the one in the last link, page 23 of Vaught Set Theory

which is extremely short & also Bourbaki's Set Theory book - but it's buried after 250+

pages of prerequisite theory. There may just be a different name for this concept, idk...

If there's a book that describes how structures on sets fall out of ZFC theory in a

book describing ZFC that would be optimal, for all I know every book does this just

under a different name.

I started a different thread a while ago trying to ground a vector space in terms of

set theory making everything very explicit, the answer I got was structured to follow

patterns that I now recognise as coming out of this idea of algebraic structures &

basically I'd just like to read how this concept is defined and originates from set theory

with all the prerequisite set theory knowledge that goes with it being built up too.

notion is not just common language!

DEFINITION: An algebraic structure consists of one or

more sets closed under one or more operations, satisfying

some axioms

http://www-public.it-sudparis.eu/~gibson/Teaching/MAT7003/L5-AlgebraicStructures.pdf [Broken]

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets

or carriers or sorts, closed under one or more operations, satisfying some axioms.

link

Then below this definition they give another (equivalent) definition:An Algebraic Structure is defined by the tuple [itex]<A, o_1, ...,o_k;R_1,...,R_m;c_1,...,c_k>[/itex]

where;

A is a non-empty set,

[itex]o_i[/itex] is a function [itex]A^{pi} \ : \ A \ \rightarrow \ A[/itex]

[itex]R_j[/itex] is a relation on A

[itex]p_i[/itex] is a positive integer

[itex]c_i[/itex] is an element of A

link (Ch. 2)

So based on this I have three questions:An algebraic structure is a triple <A,O,C> where:

A ≠ ∅

[itex]O \ = \ U^n_{i = 1} \ o_i[/itex] where o_i are i-ary operations

C ⊆ A is the constant set.

**1: How is this concept explained in terms of set theory?**I am thinking that it follows from the idea of a "structured set".

Unfortunately, I can find basically nothing on this concept from browsing online.

The only sources I have found are the one in the last link, page 23 of Vaught Set Theory

which is extremely short & also Bourbaki's Set Theory book - but it's buried after 250+

pages of prerequisite theory. There may just be a different name for this concept, idk...

**2: Could you recommend any**

discussing Structures on Sets as they arise in ZFC theory?__sources__(book recommendations)discussing Structures on Sets as they arise in ZFC theory?

If there's a book that describes how structures on sets fall out of ZFC theory in a

book describing ZFC that would be optimal, for all I know every book does this just

under a different name.

**3: Could you recommend any**__sources__explaining Algebraic Structures in terms of sets?I started a different thread a while ago trying to ground a vector space in terms of

set theory making everything very explicit, the answer I got was structured to follow

patterns that I now recognise as coming out of this idea of algebraic structures &

basically I'd just like to read how this concept is defined and originates from set theory

with all the prerequisite set theory knowledge that goes with it being built up too.

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