The study of algebraic structures and properties applying to large classes of such structures. For example, ideas from group theory and ring theory are extended and considered for structures with other signatures (systems of basic or fundamental operations).

learn more… | top users | synonyms

8
votes
1answer
279 views

Combinatorics of term algebras

My question is about the number of terms of size $n$ in term algebras for an arbitrary (finite) signature. A signature is a map $\Sigma : S \rightarrow \mathbb{N}$ from a set $S$ of symbols. We ...
1
vote
3answers
192 views

n-ary derived operation in universal algebra

I've just come across the definition of the n-ary derived operation, namely that starting with an operational type $(\Omega, \alpha)$, set $ X_n = (x_1, ... , x_n) $ and $ \Omega$-structure $A$, we ...
4
votes
1answer
203 views

Introductory universal algebra question

I've just started reading about universal algebra, and have already hit a problem (see the two bullet points at the bottom). My book gives the following definitions (paraphrased): An operational ...
1
vote
1answer
95 views

Idea for a proof involving an identity of term functions on $\sigma$-structures [x]

I have some problems with the following theorem: Fix an signature $\sigma$ and a set of variables $\mathbb{V}$. We call $t$ and $t_1$ "equivalent", if for every $\sigma$-structure $S$ and every term ...
105
votes
5answers
7k views

In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
2
votes
2answers
141 views

how are vector spaces viewed as universal algebras?

Hey I have this question from Universal Algebra texts where you can see groups, rings, lattices and other structures as Universal Algebras, but I still don't have clear how vector spaces can be viewed ...
1
vote
1answer
104 views

Count of unique algebras with one unary operation (on finite set)

It's possible, that count of unique algebras (up to isomorphism) with one unary operation on set with $n$ elements is $2^n-1$? For $n=1,2,3,4$ is this hypothesis true (I still have not verified it on ...
1
vote
1answer
154 views

Can nonisomorphic algebras generate isomorphic relatively free algebras?

Suppose that we have two varieties of algebras $A$ and $B$, whose operators all have arities less than some regular cardinal, and such that every $B$-algebra (please correct me if this is not the ...
1
vote
1answer
65 views

How to define the action of U(G) in this situation?

The usual action of $fg$ on $u⊗v$, where $f,g$ are elements in the Universal Enveloping Algebra $U(G)$ of a Lie algebra $G$ and $u,v$ are elements of a representation $V$ of $G$, is given by ...
1
vote
3answers
405 views

Can we represent all algebraic structures in First-Order logic?

Can we represent all algebraic structures in First-Order logic?
2
votes
1answer
185 views

Algebra terminology question

Suppose A is some algebraic structure, and x and y are two elements of the underlying set. Is there any more concise way of stating, "there exists an automorphism in A which maps x to y"?. It seems ...
2
votes
5answers
572 views

Constructing a counterexample in category theory

Exercise 10 in Geroch's Mathematical Physics asks whether direct products distribute over direct sums in arbitrary categories. (They do in the category of sets, which is what motivates the question). ...
26
votes
4answers
2k views

Simple explanation of a monad

I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...