# Tagged Questions

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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### Counterexamples in universal algebra

Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's ...
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### Identities of commutators

Let $G$ be a group and set $[x,y]:= x^{-1}y^{-1}xy$ as usual and consider it as a binary operation. Question: Is there a description of the identities that the operation $[.,.]$ satisfies for all ...
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### A preprint of Sela concerning the work of Kharlampovich-Miyasnikov

Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free ...
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### 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 (constants)...
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### Lawvere theories versus classical universal algebra

A Lawvere theory is a small category with finite products such that every object is isomorphic to a finite product of copies of a distinguished object x. A model of the theory in a category with ...
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### 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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### Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
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### Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
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### Why is every variety of bands determined by a single identity?

A band is a semigroup where every element is idempotent: $a a = a$. A collection of bands is a variety if it is closed under taking subobjects (subsemigroups), quotient objects (images of ...
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If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like min(min($x+... 2answers 1k views ### What is the status of (universal) algebra in type theory? With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature ... 2answers 456 views ### Contexts and notations for composing asymmetric simplices Imagine the elements of a group-like structure as puzzle pieces with essential two sides, an IN-side and an OUT-side. You can compose two such pieces in two obvious ways: Now consider triangular ... 0answers 791 views ### Why did Bourbaki not use universal algebra? I have seen a discussion about Bourbaki’s usage of categories before. So let me ask a different question: why did he not use universal algebra? Well, universal algebra is not much older than category ... 1answer 462 views ### Which categories are the categories of models of a Lawvere theory? Background: a Lawvere theory$T$is a category with finite products such that each object is a power of a fixed object$x$. Given a Lawvere theory$T$, the category$\text{Mod}_T$of models of$T$is ... 1answer 924 views ### So, did Poincaré prove PBW or not? This seems to be a question whose answer depends on whom you ask. Maybe we can come up with a final answer? It is known that Poincaré, at least, invented something that can be called Poincaré-... 1answer 285 views ### Does every commutative variety of algebras have a cogenerator? By a commutative variety$\mathcal{V}$I mean a classical variety of algebras for some$(\Sigma,E)$, such that each pair of operations in$\Sigma$commutes. Equivalently (i) every interpretation of ... 5answers 682 views ### What are the invariant Pseudo-differential operators on a Lie group? It is well-known that (left)$G$-invariant differential operators on a Lie group$G$, has an algebraic description, i.e. universal enveloping algebra of the Lie algebra of the group. On the other ... 0answers 370 views ### When can we tell if PROPs, Algebraic Theories, etc. are faithfully detected in a given category? I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am ... 2answers 822 views ### Free division rings? Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators? 3answers 632 views ### Natural associative law for a ternary “group”? Suppose one were to define a group-like structure based on a set$G$with a ternary (rather than binary) operator$g( a, b, c ) = \left< a, b, c \right>$. One possible definition for the ... 4answers 585 views ### The groupoid of algebraic expressions and proofs Fix a set of variables$V$, and suppose we're given a presentation of a monosorted algebraic theory, with variable symbols taken from$V$. For the sake of example, suppose the presentation consists of ... 3answers 526 views ### Are norms intrinsically$\mathbb{R}$-valued? Another way of phrasing this: are there any viable definitions of something which is norm-like but whose range is in a linearly ordered rig (for example) rather than$\mathbb{R}$? I have searched a ... 1answer 232 views ### Does every Lawvere theory arise in this way? By a Lawvere theory, I mean a finite-product category$\mathsf{T}$equipped with a distinguished object, such that every object of$\mathsf{T}$can be expressed as a finite product of the ... 4answers 622 views ### Complete Boolean algebra not isomorphic to a$\sigma$-algebra Does there exist a complete Boolean algebra that is not isomorphic to any$\sigma$-algebra? If so, what is an easy or canonical example or construction? 3answers 704 views ### What is the smallest variety of algebras containing all fields? A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ... 3answers 741 views ### When finitely generated free algebras are finite The variety (in the sense of universal algebra) of Boolean algebras, for example, has the property that finitely generated free algebras have finite cardinality; in that case specifically$|F_n|=2^{2^...
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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 ...
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### Algebraic axiomatization for AB+BA^T operation on matrices

Let us consider a matrix algebra $Mat_{n\times n}(K)$, where $K$ is a field, $char K \neq 2.$ It is well-known that the axiomatization of commutator operation $[A,B]=AB-BA$ on matrix algebra leads us ...
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### Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...
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### Is the word problem decidable for free finitely generated self-square groups?

A self-square group is a group with extra structure, which encodes the fact that the group is isomorphic to its own direct square. To be exact, the group $G$ has a special element $1$, a unary ...
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### When can I assure that the representation theory of a PROP is faithful?

Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A representation or algebra for a prop $P$ in a symmetric monoidal ...
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It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way ...
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### Is there a general result that theorems about finite structures proved in ZFC can be proved in ZF?

The title question is too vague so let me be specific. Much of modern finite semigroup theory uses profinite semigroups and properties of profinite semigroups that depend on the existence of prime ...
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### Primitive Recursive Arithmetic via Universal Algebra

From the wikipedia article on Primitive Recursive Arithmetic (see http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic): "Primitive recursive arithmetic, or PRA, is a quantifier-free ...
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### IBN for algebraic theories

Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
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### Is HSP(A) = ISP(A) decidable?

Let $A$ be a finite algebra for some finitary signature. Is it decidable whether $\mathbb{H}\mathbb{S}\mathbb{P}(A) = \mathbb{I}\mathbb{S}\mathbb{P}(A)$? That is, whether the variety ...
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### Varieties of groups with infinite relatively free group of rank 2 finite, infinite in rank 3

Does there exist a variety of groups $\mathfrak{V}$ such that the relatively $\mathfrak{V}$-free group of rank 2 is finite, but the relatively $\mathfrak{V}$-group of rank 3 is infinite? (In other ...
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### Fuzzy logic of Godel

In Gödel logic, is conjunction definable from implication, negation , and disjunction? We know that conjunction in that logic is not definable from negation and implication.
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Setup: Let $\mathcal{O}$ be an operad in the category of sets, and let $\mathcal{O}\text{-Alg}$ denote the category of algebras on it (i.e., operad functors $\mathcal{O}\to\mathbf{Set}$. This category ...
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### A decision problem for clones

E. Post proved that there are only countably many clones on a two-element set (classes of operations closed under superposition and containing all projections). All these clones are finitely ...
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### 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 ...