26 votes
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Are there axioms satisfied in commutative rings and distributive lattices but not satisfied in commutative semirings?

Following Fran├žois's suggestion, I ran alg to find a unital commutative semiring which fails to satisfy $$ \forall x\, y\, z,\; x + z = y + z \land x \times z = y \times z \Rightarrow x = y. \tag{1} ...
Andrej Bauer's user avatar
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26 votes
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Is there an identity between the commutative identity and the constant identity?

Yes: $$(x + x) + y = y + x$$ The constant identity implies this because both sides are $+$es. This does not imply the constant identity because it is true about any set with an operation that is ...
paste bee's user avatar
  • 1,171
22 votes

How dangerous are set-size assumptions?

For what it's worth, the reason I made that comment was because when I gave talks expressing skepticism about the philosophical basis of ZFC, I would often get the reaction "but as long as it's ...
Nik Weaver's user avatar
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22 votes
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Why is the theory of small categories not algebraic?

This follows from two Facts: 1) A category monadic over Set/S is always an exact category. That is it has quotient by equivalence relation that are effective and universal. It is in particular a ...
Simon Henry's user avatar
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20 votes
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Representation theorem for modular lattices?

There are lots of relations satisfied in lattices of submodules besides the ones implied by modularity. For example, there is the Desarguesian identity mentioned here (which holds in any lattice of ...
Todd Trimble's user avatar
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19 votes
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In what sense are operads "better" than PROPs?

I was friends with Frank Adams and Saunders Mac Lane, who invented PROPs in one of the world's most extensive unpublished collaborations. Saunders once showed me a box full of their correspondence. ...
Peter May's user avatar
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19 votes
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What is a module over a Boolean ring?

Theorem: Given $A$ a boolean ring/boolean algebra then there is an equivalence of categories between the category of $A$-modules and the category of sheaves of $\mathbb{F}_2$-vector spaces on Spec $A$....
Simon Henry's user avatar
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19 votes
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How do finite door spaces work?

A door space $X$ is $T_0$, for if $x,y\in X$ are not separated by the $T_0$ axiom then the set $\{x\}$ is neither open nor closed. A finite $T_0$ space is equivalent to a finite poset $P$ (Enumerative ...
Richard Stanley's user avatar
18 votes
Accepted

Is there an infinite chain of endofunctors of finite sets?

Yes. Let $F_i$ be the subfunctor of the covariant powerset functor given by $S\in F_i(X)$ iff $0<|S|\le i$. If $X$ has exactly $i$ elements, then there is an element of $F_i(X)$ fixed by all ...
Tom Goodwillie's user avatar
17 votes

Relation between monads, operads and algebraic theories (Again)

Lawvere theories can be thought of as "cartesian operads." That is, we have an analogy $$\text{Lawvere theories} : \text{cartesian monoidal categories} :: \text{operads} : \text{symmetric monoidal ...
Qiaochu Yuan's user avatar
16 votes

In what sense are operads "better" than PROPs?

One thing you can do with an operad that you cannot do with a prop is write down a monad such that algebras over the monad correspond to algebras over the operad. For example, Hopf algebras have a ...
Qiaochu Yuan's user avatar
16 votes
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Constructive proof of existence of free algebras for infinitary equational theories

It was proved by Andreas Blass in Words, free algebras, and coequalizers that free infinitary algebras are not constructible neither in topoi nor in ZF. It is easy to see that the existence of free ...
Valery Isaev's user avatar
  • 4,390
16 votes
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Ultrafilter subtraction and "zero"

Both your guesses are correct. To see this, it's helpful to reformulate the way you're thinking about the subtraction operator on $\beta \mathbb Z$. Beginning with subtraction on $\mathbb Z$, you can ...
Will Brian's user avatar
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15 votes
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characterization of subalgebras of universal enveloping algebra coming from Lie subalgebras

In characteristic $0$, if $Y$ is a Hopf subalgebra of $U(\mathfrak{g})$, then $Y = U(\mathfrak{g}')$ for some Lie subalgebra $\mathfrak{g}'$ of $\mathfrak{g}$ (and conversely, every such subalgebra is ...
Christopher Drupieski's user avatar
15 votes
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Generalized cancelation properties ensuring a monoid embeds into a group

The answer is no. What you call a generalized cancellation rule is called a quasi-identity in universal algebra. Malcev proved in 1939 that there is no finite basis of quasi-identities defining group ...
Benjamin Steinberg's user avatar
15 votes
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How many sublattices are contained in the powerset lattice of a finite set?

OEIS A306445: 2, 4, 13, 74, 732, 12085, 319988, 13170652, 822378267, 76359798228, 10367879036456, 2029160621690295, 565446501943834078, 221972785233309046708, 121632215040070175606989, ...
Keith Kearnes's user avatar
15 votes
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Is there an identity between the associative identity and the constant identity?

An identity $E$ that obeys all the claimed properties is $$ E: x+(y+z) = (x+y)+w \hbox{ for all } x,y,z,w.$$ $E$ is implied by triple constancy (and hence by constancy): obvious since both sides are ...
Terry Tao's user avatar
  • 107k
14 votes

Representation theorem for modular lattices?

The lattice of submodules of a module satisfies a stronger identity, namely the Arguesian law: $$ (x_0\vee y_0)\wedge (x_1\vee y_1)\wedge (x_2\vee y_2)\leq ((z\vee x_1)\wedge x_0) \vee ((z\vee y_1)\...
Pedro Sánchez Terraf's user avatar
14 votes
Accepted

Is the class of power-associative binars finitely axiomatizable?

No. Indeed, let $\mathcal{V}_n$ be the variety of magmas generated by the relating identities with variable $y$ saying that for every $k\le n$, all products of $k$ copies of $y$ are equal. Since the ...
YCor's user avatar
  • 60k
14 votes
Accepted

Is the equational theory of this "orthocentrish" algebra finitely based?

This algebra is finitely based. In fact, if you choose any bijection from $\{a,b,c,d\}$ to $\mathbb Z_2\times \mathbb Z_2$, then you can transport the operation $F(x,y,z)$ to $\mathbb Z_2\times \...
Keith Kearnes's user avatar
14 votes
Accepted

Are modular lattices shallow?

Terminology. If $f$ is a fundamental operation of an algebra $A$, then a polynomial of the form $f(c_1,\ldots,c_k,x,c_{k+2},\ldots,c_n)$ is often called a 'basic translation' or a '$1$-translation' ...
Keith Kearnes's user avatar
13 votes
Accepted

What is known about ideal and divisibility lattices of GCD domains and their generalizations?

Given a ring $R$, let us denote by $L(R)$ the lattice of two-sided ideals of $R$ for which the infimum and supremum are given by $\inf(I, J) = I \cap J$ and $\sup(I, J) = I + J$. Such lattices are ...
Luc Guyot's user avatar
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13 votes
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Can a Shelah semigroup be commutative?

An infinite Shelah semigroup must be a Jonsson semigroup (meaning that it is an infinite semigroup whose proper subsemigroups have lesser power). Therefore the following paper answers the question ...
Keith Kearnes's user avatar
13 votes

What classes of groups can arise as "symmetry groups of terms"?

Let me edit this response in order to clarify what I am showing. First, I will begin with an example: $t(x_1,x_2,x_3,x_4,x_5,x_6) = ((((x_1x_2)x_3^{-1})x_3)x_4)x_4^{-1}$ is a group term. If you want ...
Keith Kearnes's user avatar
13 votes
Accepted

What algebraic structure controls endomorphisms of algebras over a Lawvere theory

Yes, for any Lawvere theory $\mathbb{T}$, there is a Lawvere theory classifying the endomorphisms of $\mathbb{T}$-algebras. This can be constructed more generally as a tensor product of Lawvere ...
Rafaël Bocquet's user avatar
12 votes

Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)

Since it has not been mentioned yet here (in this question- certainly it appears somewhere on MathOF), let me mentioned that such a phenomenon holds in a considerably larger setting. An object $K$ of ...
YCor's user avatar
  • 60k
12 votes

Ternary associative multiplication

Perhaps this has already been said in some form, but the identity $$[[a,b,c],d,e] = [a, [b,d,e], [c,d,e]]$$ is exactly the $(2,2,2)$-associative law of clone theory. I think that this is the most ...
Keith Kearnes's user avatar
12 votes
Accepted

How dangerous are set-size assumptions?

This answer repeats some of the material in other answers but I think it is not entirely redundant. As you seem to have realized, the assertion that a theory is consistent is a much weaker statement ...
Timothy Chow's user avatar
  • 77.9k

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