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78
votes
5answers
4k 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 ...
22
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 ...
15
votes
2answers
403 views

A structural proof that $ax=xa$ forms a monoid

During the discussion on this problem I found the following simple observation: If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid. This is trivial to prove by checking ...
9
votes
4answers
404 views

Are isomorphic structures really indistinguishable?

I always believed that in two isomorphic structures what you could tell for the one you would tell for the other... is this true? I mean, I've heard about structures that are isomorphic but different ...
9
votes
2answers
118 views

Suggestions for a learning roadmap for universal algebra?

I think a useful combination of resources for universal algebra would ideally, when taken together: Provide ample motivation behind the various developments in the field. Either provide powerful ...
9
votes
1answer
219 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 ...
8
votes
1answer
268 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 ...
8
votes
1answer
208 views

An exercise in infinite combinatorics from Burris and Sankappanavar

Exercise 6.7 in chapter IV of Burris and Sankappanavar's A Course in Universal Algebra starts as follows: Show that for $I$ countably infinite there is a subset $S$ of the set of functions from ...
7
votes
1answer
193 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 ...
6
votes
2answers
97 views

In a slice category C/A of a category C over a given object A, What is the role of the identity morphism of A in C with respect to C/A

In a slice category $C/A$ of a category $C$ over a given object $A$, what is the role of the $C$ identity morphism, $A\to A$ ($1_A$), in $C/A$, particularly with respect to composition? I ...
6
votes
4answers
165 views

Any commutative associative operation can be extended to a function on nonempty finite sets

This is a fact we use very frequently in general mathematics when we write such notations as $1+2+3+4$: since we know that $+$ is commutative and associative, we can just "drop the parentheses" and ...
6
votes
1answer
76 views

Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
6
votes
1answer
110 views

What is a simple axiomatisation of groups using division?

I recall from an old exercise I did as an undergrad that groups can be axiomatised using division rather than multiplication: A group is a non-empty set equipped with a binary division operator / ...
5
votes
1answer
172 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 ...
5
votes
1answer
92 views

$\mathbf{N}_5$ as a congruence lattice

A finite lattice is said to be representable if there exists a finite algebra whose congruence lattice is isomorphic to that lattice. As I was reading a paper, I came across the line: "The reader can ...
5
votes
0answers
37 views

Is there a name for the algebra of substructures?

Let $X$ denote an entropic algebra (see here), which just means that all the operations of $X$ are homomorphisms $X^n \rightarrow X.$ Abelian groups are the classic example. Then for any operation of ...
4
votes
1answer
85 views

How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...
4
votes
1answer
45 views

Which sentences survive the passage from $X$ to the set of all functions $I \rightarrow X$?

Suppose $X$ is a mathematical structure with a single underlying set which we will also denote $X$, equipped with some functions and relations. Letting $I$ denote an arbitrary non-empty set, we see ...
4
votes
2answers
148 views

Stone duality for ideals and filters (exercise)

In A Course in Universal Algebra (Burris, Sankapannavar), the exercise 4.4.7-8, p.158, says: Let $A$ be a Boolean algebra. Denote $A^\ast:=\{\text{ultrafilters of }A\}$, and give $A^\ast$ the ...
4
votes
1answer
73 views

The function $f(x)=(x\vee a)\wedge b$ in a lattice.

Is there an algebraic modular lattice $(X,\vee,\wedge)$ and $a,b\in X$ with $a\le b$ such that the function $$f:X\to X$$ $$f(x)=(x\vee a)\wedge b$$ is not $\vee$-homomorphism?
4
votes
1answer
63 views

Associativity, Jacobi, and self-action representations

About a year and a half ago, I was at a talk by Martin Hyland where he suggested that the Jacobi identity is to the associative law as the anticommutative law is to the commutative law. I think this ...
4
votes
1answer
151 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 ...
4
votes
2answers
222 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 ...
4
votes
1answer
73 views

The “closed” subspaces of topological algebraic structures

Every set-theoretic model of an algebraic theory gives rise to notion of (algebraically) "closed subset" in a canonical fashion; namely, the closed subsets are those that cannot be escaped via the ...
3
votes
3answers
310 views

New kind of identities?

I found a new kind of identities which are half logic and half algebraic while working on a proof of NP-completeness. They are like this: $$ \frac{a+mb}{n+m} < \frac{a}{n} \iff b < ...
3
votes
2answers
253 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, ...
3
votes
2answers
209 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
votes
1answer
37 views

Confirmation needed of the fact that subcategory $\mathbf{Lat}$ is not full in $\mathbf{Pos}$

If you are familiar with this stuff then you probably don't need the information I have added. So let me start with the question: Can you prove that category $\mathbf{Lat}$ is not a full ...
3
votes
2answers
44 views

Why are the algebras of the associative operad unital?

According to the n-lab page: The associative operad Assoc is an operad which is generated by a binary operation $\Theta$ satisfying $$\Theta\circ(\Theta,1)=\Theta\circ(1,\Theta)$$ It then ...
3
votes
1answer
136 views

Do filters on a Boolean algebra also make a Boolean algebra?

Let $\mathfrak{B}=(B,\bot,\top,\lnot,\wedge,\vee)$ be a boolean algebra. $B_F$ be the set of all filters on $\mathfrak B$. And for all filter $F$, $G$, $F \wedge_{B_F} G \colon= \mathbf C(F \cup G)$ ...
3
votes
2answers
70 views

For what categories do category algebras exist?

The monoid algebra $R[M]$, for a commutative ring $R$ and a monoid $M$, can be described as the free $R$-algebra on $M$. We think of $R[M]$ as the set of finite formal sums of elements of $M$ with ...
3
votes
2answers
76 views

Categories without identities

What's the name of "categories without identities", i.e. of digraphs with just an associative binary operation on its "matching" arrows (disregarding identities)?
3
votes
1answer
116 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' ...
3
votes
1answer
105 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 $F$ ...
3
votes
1answer
24 views

Signatures having precisely one constant symbol, and pointed categories.

Given an algebraic signature $\sigma$ having precisely one constant symbol, is it true that if $A$ is a set of quasi-identities in the language of $\sigma$, then the set-theoretic models of ...
3
votes
1answer
76 views

$\mathrm{Pol}_m(\mathbb{A})$ viewed as a relation pp-definable from $\mathbb{A}$

First let me recall some (abbreviated, and possibly simplified to suit my situation) definitions: Let $A$ be a finite set and $\mathbb{A}$ some set of relations on $A$. Let $m, n$ be positive ...
3
votes
0answers
44 views

Seeking information about (category-theoretic) varieties, quasivarieties, and universal Horn classes.

I'm looking for a list of basic facts regarding (category-theoretic) varieties, quasivarieties, and universal Horn categories, as well as information about which forgetful functors preserve what. In ...
3
votes
0answers
83 views

What big families of theories/structures are there?

To gain a big picture of (pre-categorical) mathematics, is it correct to divide mathematical theories resp. structures in two big families? universal algebra: classes of objects with arbitrary ...
2
votes
4answers
209 views

why is a nullary operation a special element, usually 0 or 1?

Does a nullary operation mean an operation not taking any argument? Then why is a nullary operation a special element, usually 0 or 1, in an algebraic structure? Thanks!
2
votes
1answer
546 views

Modus Ponens: implication versus entailment

Would it be inconsistent to write Modus Ponens using only implication, not entailment? $(p \wedge (p \to q)) \to q$ The way I understand is that implication ($ \to$) is an operator that yields a new ...
2
votes
3answers
132 views

Preserving structures

Category theory abstracts the notion of the preservation of structure by means of morphisms. Is there a description of what it means to preserve structure of different types of mathematical structures ...
2
votes
3answers
163 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 ...
2
votes
2answers
112 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
2
votes
2answers
40 views

Class of algebras defined by quasi-identities is not closed under taking the quotients

Let $\mathbf{A}$ be an algebra (in the sense of universal algebra) of some signature $\Sigma$. By quasi-identity I mean the formula of the form $$(\forall x_1) (\forall x_2) \dots (\forall x_n) ...
2
votes
2answers
164 views

Isomorphism of algebras $\mathbb{Q}(\cdot , +, 1)$ and $\mathbb{Z}(\cdot , +, 1)$

I have these two algebras and I need to know if they are isomorphic: $\mathbb{Q}(\cdot , +, 1)$ and $\mathbb{Z}(\cdot , +, 1)$ Are there some general tricks how to deal with this type of tasks?
2
votes
3answers
152 views

Free object is a coproduct: $F_{A\cup B}\cong F_A \coprod F_B$

Let $A,B$ be sets, and $A\sqcup B$ the disjoint union. Suppose that in a (concrete) category, the free objects $F_A,F_B,F_{A\sqcup B}$ exist, and that the coproduct $F_A \coprod F_B$ exists. How can I ...
2
votes
2answers
51 views

Do formal polynomials make sense in arbitrary algebraic structures?

Let $R$ denote a commutative ring with unity and $X$ a set of formal indeterminates. Then intuitively, the set of all formal polynomials in $X$ with coefficients in $R$ can be defined as the free ...
2
votes
2answers
121 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 ...
2
votes
1answer
72 views

Does axiomatizability in zeroth-order logic have important consequences?

If a theory is equationally axiomatizable, this has important consequences (that are studied e.g. in universal algebra). However, many theories fail to be equationally axiomatizable - examples ...
2
votes
1answer
104 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 ...