The universal-algebra tag has no wiki summary.

**6**

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519 views

### Binary operations compatible with the usual order on the reals

An officemate passes along the following natural-seeming question:
Say that a binary operation $\oplus$ is compatible with the usual order $\leq$ on $\mathbb{R}$ if for any $w, x, y, z$ in ...

**22**

votes

**20**answers

2k views

### Why are so few operations with arity bigger than 2?

In usual algebraic structures, like groups, rings, monoids, etc, or in algebras coming from logics like Boolean algebras, Heyting algebras and the like the operations are usually of arity 0 ...

**7**

votes

**1**answer

579 views

### Lattice-ordered commutative monoids

By a lattice-ordered monoid, I mean a structure $(A,0,{+},{\vee},{\wedge})$ such that $(A,0,{+})$ is a (not necessarily commutative) monoid, $(A,{\vee},{\wedge})$ is a lattice, and the two ...

**3**

votes

**2**answers

545 views

### Is the decomposition of an algebra into irreducible components essentially unique?

We consider finite algebras for a given signature, in the sense of universal algebra (for example, they might be groups, rings, or lattices). An algebra $A$ is irreducible when $A \cong B \times C$ ...

**3**

votes

**2**answers

236 views

### What do you call a lattice whose meet operation preserves disjointness of subsets?

To make my question more precise and compact (and probably more intuitive), let me define the following:
A subset $S$ of a lattice is mutually disjoint if for each $x \in S$, $\bigvee(S - \lbrace x ...

**0**

votes

**1**answer

147 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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**0**answers

518 views

### Counting and understanging commuting functions.

Fix a positive integer $n$, and consider the functions from a set of size $n$
to itself. Let $cp(n)$ denote the number of ordered pairs $\langle f,g
\rangle$ of these functions which commute, i.e., ...

**4**

votes

**2**answers

243 views

### Theorems (from clone theory) that can be stated only by using operations and their composition.

To state my question, I first have to explain what I mean by "using operations and their compositions only". In short, it means that I am looking for theorems from clone theory (or on operations and ...

**2**

votes

**1**answer

321 views

### Essential arities in quasi-varieties.

For my question, let us consider the following scenario.
We have a quasi-variety $\mathcal{A}$ generated by a finite algebra $\mathbf{M}$ (i.e. $\mathcal{A} = \mathbb{ISP}(\mathbf{M})$). Now, let ...

**6**

votes

**5**answers

479 views

### Computing colimits in a Lawvere theory

Let Prod(C, D) be the set of finite-product preserving functors from C to D. Is it true that for any Lawvere theory L, Prod(L, Set) has small colimits? This seems to be the case, as it is invoked ...

**4**

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**2**answers

566 views

### Is there a notion of congruence relation for essentially algebraic structures?

In universal algebra there is the notion of congruence relation: Consider a (1-sorted) algebraic structure, i.e. a set $A$ with a bunch of finitary operations $f_i$ satisfying equations.
A congruence ...

**1**

vote

**4**answers

760 views

### Is monomorphism going in both directions sufficient for isomorphism?

In category theory, it seems that a monomorphism from $A$ to $B$ and one from $B$ to $A$ should be enough to guarantee isomorphy, but it doesn't seem to be so. (If I'm right then there's something ...

**4**

votes

**0**answers

260 views

### The graph of algebraic theories

Fix a logic $L$ and consider the category $\mathbf{AlgTh}_L$ with theories of $L$ as objects and theory interpretations as morphisms. For nice enough logics, this category has pushouts (which we will ...

**1**

vote

**1**answer

195 views

### What are sources for pathological and non-so-pathological Gabriel filters on commutative rings?

The heavy lifting in the theory of Gabriel filters is for noncommutative rings, and discussions I've been able to find all focus there.
I am trying to develop a theory of Gabriel-filter localization ...

**9**

votes

**2**answers

743 views

### Free division rings?

Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators?

**17**

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**6**answers

1k views

### Do non-associative objects have a natural notion of representation?

A magma is a set $M$ equipped with a binary operation $* : M \times M \to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional ...

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vote

**2**answers

841 views

### What is a semigroup or, what do I do with that associativity proof?

Mathematically, I know what a semigroup is: It is a set S along with an associative binary operation $* : S \times S \rightarrow S$. So far, so good.
From a computational perspective, one can ...

**8**

votes

**2**answers

608 views

### Right actions of operads and monads

Given an operad $A$, there is an associated monad $M_A$ given by $M_A(X) = \coprod_n A(n) \otimes X^{\otimes n}$ such that being an $A$-algebra and being an algebra over the monad $M_A$ is the same ...

**2**

votes

**1**answer

453 views

### Higher-order, multi-sorted, non purely equational version of universal algebra ?

I have been looking around, unsuccessfully, for generalizations of universal algebra based on higher-order logic (rather than first order) and where the relations are not purely equational. ...

**3**

votes

**5**answers

894 views

### Defining 'free monoid' without Nat?

Is there a definition of what is a 'free monoid' which does not pre-suppose that the natural numbers has already been defined? The definitions that I have been able to track down all use the natural ...

**3**

votes

**1**answer

176 views

### Equational definition of Residuated Lattices

The usual axiomatization of residuated lattices involve using ≤. I know I can expand away the use of ≤ using a definition such as (x ≤ y) := (x ∧ y = x), but I fear I will get a set of messy axioms.
...

**3**

votes

**2**answers

265 views

### Terminology: Name for a homomorphism from the free object?

Is there a standard name for taking a homomorphism from the free object over an algebraic structure? Roughly speaking, this should amount to evaluation of any element of the free object under the ...

**1**

vote

**1**answer

485 views

### Coequalizer in the category of primitive recursive functions

What does a coequalizer in the category of primitive recursive functions look like? I know that in Set, a coequalizer is a minimum congruence but...what is it in particular in the category of ...

**7**

votes

**1**answer

489 views

### Does ⋄ generate all De Morgan algebras?

(Asked by Nathaniel Hellerstein on the Q&A board at JMM)
This question is about De Morgan algebras (see also Wikipedia), which are something like Boolean algebras, but with a different weaker ...

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votes

**2**answers

424 views

### Is every lattice the fixed-point set of an order endomorphism of ⋄^n?

(Asked by Nathaniel Hellerstein on the Q&A board at JMM)
Let ⋄ be the 4 element lattice
τ
/ \
i j
\ /
f
Is every lattice isomorphic to the fixed point lattice of some ...