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).

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150 views

correspondence for universal subalgebras of $U/\vartheta$

Let $U$ be a universal algebra of type $T$, and denote $\mathrm{Con}(U)\!=\!\{\text{congruence relations on }U\}$ and $\mathrm{Sub}(U)\!=\!\{\text{subalgebras of }U\}$. Let "$\leq$" mean "subalgebra". ...
3
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2answers
263 views

What is the name of the structure Z4 under subtraction?

If we consider $\mathbb{Z_4}$ under addition, then it forms a cyclic group of order 4. However if we change the binary operation to subtraction on $\mathbb{Z_4}$, we get a different structure $J$ with ...
3
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1answer
173 views

$M_3$ is a simple lattice

I'd like to prove (exercise 9.5 in Roman's Lattices and Ordered Sets, p.203) that the lattice $M_3$ is simple, meaning that the only congruences on $M_3$ are the trivial ones (the 'equality' ...
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1answer
145 views

ideals of a ring form a modular lattice

We know that if $M$ is a left $R$-module, then $(\{\text{submodules of }M\},\subseteq)$ is a modular lattice. Taking $M\!=\!R$, we deduce that $(\{\text{ideals of }R\},\subseteq)$ is a modular ...
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2answers
305 views

Variety generated by finite fields

Let $K_1,\dotsc,K_n$ be finite fields and let $V$ be the variety of rings, generated by the $K_i$ (rings aren't necessarily unital). I want to figure out what $V$ looks like. By a theorem of Tarski, ...
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279 views

Lattices are congruence-distributive

$\newcommand{\r}[1]{\mathrel{#1}}$ First, a few definitions. Given a lattice $L$, a congruence on $L$ is an equivalence relation $\theta$, compatible with the lattice operations, i.e. if ...
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3answers
239 views

Are groups algebras over an operad?

I'm trying to understand a little bit about operads. I think I understand that monoids are algebras over the associative operad in sets, but can groups be realised as algebras over some operad? In ...
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2answers
258 views

What is the smallest variety of algebras containing all fields?

A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
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0answers
85 views

direct product of different algebras?

Is it possible to define a "direct product" of two algebras with different signatures? For example, boolean lattice $\otimes$ monoid? Perhaps we need to take some quotient to make sense of it (e.g Q ...
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0answers
160 views

amalgam of structures

Trying to refine my question here. This is a response to the questions here: Homomorphisms between structures My objective is to take a set of $S-$structures and form an amalgam object out of that ...
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295 views

Why do free monoids have a “trivial” automorphism group and free groups don't?

Let $X$ be a set and $M$ the free monoid over $X$. Then an automorphism $f$ of $M$ satisfies $f(X)=X$ and so $\text{Aut}(M)$ is canonically isomorphic to $\mathfrak{S}_X$. My Proof: For every word ...
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350 views

Injective Homomorphism and direct products

This question relates to products of structures all with the same symbol set $S$. After I give a little background the question follows. Direct Products This definition of the direct product is ...
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230 views

Existence of universal enveloping inverse semigroup (similar to “Grothendieck group”)

Context In its simplest form, the Grothendieck group construction associates an abelian group to a commutative semigroup in a "universal way". Now I'm interested in the following nilpotent ...
3
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1answer
130 views

Is the rank of a relatively free group… ill-defined in general?

A relatively free algebra $F$ has a free generating set (basis) $X$ such that any map $f : X \to F$ can be extended to an endomorphism of $F$. It is known that, in general the notion of rank of $F$ ...
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218 views

Software for some universal algebra issues

I am looking for some mathematical software that can help me with a very common task in the realm of universal algebra (as far as I know programs like prover9/mace4 and uacalc do not help with this ...
8
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1answer
284 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 ...
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3answers
203 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 ...
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1answer
209 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 ...
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1answer
97 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 ...
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6answers
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
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2answers
146 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 ...
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1answer
107 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 ...
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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 ...
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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 ...
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3answers
407 views

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

Can we represent all algebraic structures in First-Order logic?
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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 ...
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5answers
588 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). ...
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4answers
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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 ...