# Algebraic Structures on Sets?

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

An Algebraic Structure is defined by the tuple $<A, o_1, ...,o_k;R_1,...,R_m;c_1,...,c_k>$
where;
A is a non-empty set,
$o_i$ is a function $A^{pi} \ : \ A \ \rightarrow \ A$

$R_j$ is a relation on A

$p_i$ is a positive integer

$c_i$ is an element of A

Then below this definition they give another (equivalent) definition:

An algebraic structure is a triple <A,O,C> where:

A ≠ ∅

$O \ = \ U^n_{i = 1} \ o_i$ where o_i are i-ary operations

C ⊆ A is the constant set.

So based on this I have three questions:

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 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.

Last edited by a moderator:

## Answers and Replies

Staff Emeritus
Gold Member
Two good keywords:
• Universal Algebra
• Model Theory

Staff Emeritus
Homework Helper
This is probably not what you're looking for, but these references are the closest I get to answering your problem:

1) http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html is a very good introduction to algebraic structures.

2) http://www.math.uiuc.edu/~vddries/410notes/main.dvi treats algebraic structures from a more fundamental point-of-view. This is probably what you want, but it's quite advanced. The theory you want is in section 2.3...

Exactly what I was hoping for! :surprised Cheers/Ty/Slainte/Salute/Nostrovia! 