How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,…,n\}$ and $*$ is the unique operation on $S_{n}$ where $n*x=x$ $x*1=x+1\,\text{mod}\, n$ and if $y<n$, then $x*(y+1)=(x*y)*(x*1)$. The algebra $(S_{n},*)$ satisfies the self-distributivity law $x*(y*z)=(x*y)*(x*z)$ if and only if $n$ is a power of $2$, and if $n$ is a power of $2$, … Read more

Poincaré-Birkhoff-Witt theorem for Leibniz algebras

Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras to Leibniz algebras. Here I want to ask you about the Poincaré-Birkhoff-Witt (PBW) theorem for free Leibniz algebras? Does the PBW theorem allow … Read more

What is important about Top\mathsf{Top} with regard to models of a homological variety?

Originally from MSE. Consider an algebraic theory whose category of models in Set is homological. This is equivalent to saying that the theory has a unique constant 0 and there exists an n≥1 such that: there are n binary operations αi such that ai(x,x)=0 there is a n+1-ary operation θ with θ(α1(x,y),…,αn(x,y),y)=x Then the category … Read more

Deciding equality in free models of a (generalized) Lawvere theory

Let $F : \mathcal{C} \rightarrow \mathcal{D}$ be functor of Lawvere theories $\mathcal{C}, \mathcal{D}$ (i.e. cartesian categories where every object is isomorphic to some power of a chosen object) that preserves finite limits and the chosen object. Precomposition $- \circ F$ turns every $\mathcal{D}$-model $A : \mathcal{D} \rightarrow \underline{\text{Set}}$ into a $\mathcal{D}$-model $A \circ F : … Read more

How many compatible linear orders exist on the classical Laver tables?

Let An be the unique algebra ({1,…,2n},∗n) such that x*_{n}1=x+1\mod 2^{n} and x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z) for all x,y,z. We say that a linear ordering \preceq on \{1,\dots,2^{n}\} is compatible with A_{n} if y\preceq z\Rightarrow x*_{n}y\preceq x*_{n}z. What are some lower bounds and upper bounds of the value t_{n}? I am more interested in lower bounds than in … Read more

Finite generation of vector identities

This question is partially motivated by, although that question may not be appropriate for MO. Consider the set E of all valid equalities (without parameters) over the two-sorted structure (R,R3) in the language consisting of symbols for: the arithmetic operations addition, subtraction, and multiplication on R; the vector operations of vector addition, vector subtraction, … Read more

Properties of category of (simplicial) algebraic objects

Given a category, one is often interested in the category of (abelian) group and (commutative) ring objects in it. I would like to know what exactness properties such categories and their simplicial analogues have, e.g what can be said about the simplicial abelian groups and commutative ring objects in some category C depending on the … Read more

Why the axiomatic rank of the variety of groups is equal to three?

I am thankful of Anton Klyachko who introduced axiomatic rank to me: the axiomatic rank of a variety is the minimum number of variables which we need to define that variety by identities. It seems clear that the axiomatic rank of the variety of groups is equal to three. But why? Answer Let V be … Read more

Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,…,x_n)$ and $t'(x_1,…,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms without variables) $c_1$, …, $c_n$ the identity $t(c_1,…,c_n)=t'(c_1,…,c_n)$ follows from (the identities of) $T$ then so does $t(x_1,…,x_n)=t'(x_1,…,x_n)$. In algebraic terms this means to … Read more