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3
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1answer
37 views

In general algebra, is every generating set equipotent to a finite basis itself a basis?

Question. Let $T$ denote an algebraic theory, and suppose $X$ is the $T$-algebra freely generated by a finite set $F \subseteq X$. Suppose $G \subseteq X$ also generates $X$ and that $|G|=|F|$. ...
1
vote
1answer
45 views

Name for the embedding property

There is an exercise in Burris and Sankappanavar's "A Course in Universal Algebra": Problem: Find two algebras $\mathbf{A}_1$, $\mathbf{A}_2$ such that neither can be embedded in $\mathbf{A}_1 \times ...
2
votes
1answer
28 views

Elementary equivalence of models

I'm quite new to model theory, so please correct me if I'm using wrong terminology. I need help with an exercise from Smirnov's book "Varieties of algebras" (In Russian). Problem: Assume that a ...
1
vote
1answer
58 views

Are algebraic structures required to satisfy axioms?

Is it a requirement for algebraic structures, when studying universal algebra, to satisfy axioms? The reason I ask is because algebraic structures are only defined by a underlying set, a signature, ...
0
votes
1answer
21 views

The lattice of closed subsets of an algebraic closure operator is an algebraic lattice

Let $A$ be a set. Let $C:Su(A)\longrightarrow Su(A)$ be a function, where $Su(A)$ denotes the set of all subsets of $A$. Suppose that 1) $X\subseteq C(X)$ 2) $X\subseteq Y\rightarrow C(X)\subseteq ...
1
vote
1answer
39 views

Distributive lattices and Birkhoff theorem

I am trying to prove the teorem (Birkhoff) $L$ is a nondistributive lattice iff $M_5$ or $N_5$ can be embedded into $L$ The only part of the proof which I can't understand is this (I am copying from ...
0
votes
0answers
38 views

What do we call those functions that can be obtained from term operations by partial evaluation?

Let $T$ denote an algebraic theory and suppose $X$ is a $T$-algebra. Then a term operation of $X$ is a function $f : X^n \rightarrow X$ that is definable by an expression in the language of $T$. ...
1
vote
1answer
27 views

What must we require of an algebraic theory for the inclusion of generators map to be injective?

Let $T$ denote an algebraic theory. Then given a free $T$-algebra $F(k)$, the inclusion of generators map $\eta : k \rightarrow U(F(k))$ is usually injective, in practice. It doesn't have to be, ...
3
votes
1answer
45 views

Help defining the $\mathrm{supp}$ function on free algebraic structures.

Given an algebraic structure $F(K)$ freely generated by a set $K$ with underlying set $U(F(K))$, I'm trying to define a "support" map $\mathrm{supp} : U(F(K)) \rightarrow \mathcal{P}_\mathrm{fin}(K).$ ...
2
votes
1answer
41 views

If a finite $T$-algebra only satisfies the identities of $T$ and no others, is it a free object?

My original question was the following: Let $T$ denote an algebraic theory and suppose $X$ is a $T$-algebra. If for every identity $\eta$ in the language of $T$ we have that $(X \models \eta) ...
3
votes
1answer
135 views

Varieties of groupoids which aren't definitionally equivalent

Here is the exercise from Smirnov's book "Varieties of algebras" (in Russian). Problem: Let $\mathcal{U}$ be the variety of all groupoids $(A, \cdot)$ and $\mathcal{V}$ be the variety of all ...
3
votes
1answer
121 views

Weak Amalgamation Property for Boolean algebras

I'm trying to study universal algebra and lattice theory by myself. Just got stuck with an exercise from Gratzer's "General Lattice Theory" and it seems to me that I don't fully understand the notion ...
3
votes
1answer
41 views

A question about commutative algebraic theories and free elements on one generator

Let $T$ denote a commutative algebraic theory with a constant symbol. (We definitely need to assume that $T$ has a constant symbol, otherwise the algebraic theory of idempotent Abelian semigroups is ...
2
votes
2answers
101 views

On the category of Sets as an example of an algebraic category

What follows comes from Algebraic Theories, pag. 7. Definition An algebraic theory is a small category $\mathcal{T}$ with finite products. An algebra for the theory $\mathcal{T}$ is a functor ...
4
votes
2answers
140 views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
2
votes
1answer
49 views

“Elements” of algebraic structures

Let $T$ denote an algebraic theory, and $e$ denote the $T$-algebra freely generated by a singleton set, and write $U$ for the forgetful functor. Now suppose we're given a $T$-algebra $A.$ We might say ...
2
votes
1answer
36 views

Is there a phrase to describe those objects of $\mathbf{C}$ that can be expressed as quotients of the algebra freely generated by $X$?

Let $\mathbf{C}$ denote the category of models of an algebraic theory in $\mathbf{Set}.$ Now suppose $X$ is an object of $\mathbf{Set}$. Is there a traditional phrase used to describe those objects of ...
4
votes
1answer
73 views

Difference between abstract algebra and universal algebra

Wikipedia give this answer "Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic ...
2
votes
1answer
103 views

Can I define a category as a monoid with partially defined multiplication?

A groupoid can either be thought of as a category whose morphisms are isomorphisms, or as a generalization of a group whose multiplication is only partially defined. Can I do a similar thing with ...
3
votes
1answer
82 views

If $\mathbb{Z}$ satisfies an identity $\eta$, then every **commutative** ring satisfies $\eta$? And related questions.

Assume all rings have unity and that ring homomorphisms preserve unity. Now by general principles, if every free object in the category of rings satisfies an identity $\eta$, then every object in the ...
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votes
0answers
23 views

Terminology in universal algbera

(Fix throughout a functional language $\Sigma$.) Given an algebra $A$ with underlying set $\vert A\vert$, there is an obvious surjective homomorphism from $A$ to the free algebra generated by $\vert ...
9
votes
2answers
147 views

What kind of object is the kernel of a ring homomorphism?

The category $\mathbf{Grp}$ of groups has a zero object, namely the trivial group $1$. Since $\mathbf{Grp}$ is furthermore complete, we have the notion of a kernel of a group homomorphism. The kernel ...
1
vote
1answer
98 views

Is there any relevance between Boolean Algebras and Fields?

In some sense Boolean Algebras and Fields have same operators and constants. In both structures there are operators addition ($+$ , $\vee$), multiplication ($\times$ , $\wedge$), inverse with respect ...
2
votes
0answers
55 views

Algeraic theory and definition of multiplication: tensor product v.s. Cartesian product

In a monoidal category, one can define multiplication on a object $M$ as a morphism \begin{equation} M\otimes M\longrightarrow M, \end{equation} or a morphism \begin{equation} M\times ...
1
vote
1answer
29 views

Coproduct of $(0,1)$-Algebras

I am trying to find the coproduct of $(\mathbb {Z},0,+1) $ with itself in the category of $(0,1) $-Algebras. Finding $\mathbb {N}\sqcup\mathbb {N} $ was easy, since $\mathbb{N} $ is initial. But I ...
11
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2answers
133 views

Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...
1
vote
1answer
34 views

Determining Objects in a Semicategory

Suppose $S$ is a small semicategory (or semigroupoid, if that's your preferred term) and $\cdot$ is the binary operation on $S$. Implicit in this definition is the set $\operatorname{Ob}(S)$ and two ...
4
votes
4answers
479 views

A doubt in Bergman's notes

On pg. 8 of these notes, Bergman says that a group $G$ contains an inverse operation $i:G\to G$, along with $\mu:G\times G\to G$ and a "neutral element" $e$. Hence, a group should be referred to as ...
12
votes
3answers
122 views

Embeddings $A → B → A$, but $A \not\cong B$?

Are there any nice examples of structures (groups, modules, rings, fields) $A$ and $B$ such that there are embeddings $A → B → A$ while $A \not\cong B$? I would especially like to see an example for ...
0
votes
1answer
16 views

Reducing a set of generators to a free base

Given an algebraic structure $ A $ which is free on some subset of its underlying set, does every generating subset of $ A $ contain a free generating set? For vector spaces, this is true, what about ...
3
votes
2answers
84 views

Examples of Stone algebras which are not Boolean algebras

Grätzer, in his Lattice Theory: Foundation, describes a Stone algebra as a distributive lattice with pseudocomplementation $L$ which satisfies the Stone identity: for every $a \in L$, $\neg a \vee ...
1
vote
2answers
46 views

When do DeMorgan's laws hold in a Heyting algebra

I'm working a bit with Heyting algebras (which are pseudocomplemented distributive lattives, right?) and I have a question about DeMorgan's laws. I know that, in general, it's not the case that $-(X ...
0
votes
0answers
27 views

Congruence lattice of N5

I calculated the the congruence lattice of $N_5$ using hit and trial and then verified it with Universal Algebra calculator. But I need to prove that it is the congruence lattice of $N_5$ How should I ...
1
vote
1answer
58 views

Textbook question on variety

Suppose a variety V is defined by an infinite minimal set of identities. Show that V is a subvariety of at least continuum many varieties.
1
vote
1answer
41 views

Variety satisfying an identity.

$V$ is a variety of commutative semigroup satisfying the identity $x^2 = x^3$. I need to prove that: $|F_V(\{x_1,\dots,x_n\})|$ = $3^n -1$. Any hints on this ? $F_V$ is V-free algebra.
3
votes
1answer
103 views

Semilattices are congruence-semi-distributive

A semilattice $(S,\cdot)$ is a commutative idempotent semigroup. A congruence on a semilattice is an equivalence relation that preserves multiplication, i.e. $x_1\mathrel{\theta} y_1$ and ...
1
vote
1answer
51 views

Factor congruences of non trivial Lattices

A pair of congruences $\theta$ and $\theta^*$ are called factor congruences if $\theta \vee \theta^*$ = full congruence. $\nabla$ $\theta \wedge \theta^*$ = trivial congruence. $\triangle$ I need ...
0
votes
0answers
23 views

How does topological dense subgroup induces properties in the larger group?

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
1
vote
0answers
23 views

Question on HSP and SHPS inquality.

In the screenshots attached above George Bergman outlines his way of proving $HSP \ne SHPS$ I understand the first definition as the group of affine transformations and each element of the group ...
3
votes
2answers
57 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 ...
0
votes
0answers
42 views

H P S class operators and their inequalities

First few definitions: $A \in I(K)$ iff $A$ is isomorphic to some member of $K$ $A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some ...
2
votes
1answer
54 views

Inequality of Class operators H S and P

First few definitions: $A \in I(K)$ iff $A$ is isomorphic to some member of $K$ $A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some ...
0
votes
0answers
58 views

What do we call functions that are definable by expressions?

Let $X$ denote a model of an algebraic theory $T$. What do we call the functions $f : X^n \rightarrow X$ that are definable by some expression in the language of $T$? e.g. If $S_3$ is the symmetric ...
0
votes
1answer
50 views

Clarification about the definition of term algebras

The following definition has been given in this article. A term algebra is an algebra $ \langle \mathcal{S}, \mathcal{G} \rangle $ where every time that $g_\alpha, g_\beta \in \mathcal{G}$ and $$ ...
0
votes
1answer
36 views

Operator generating subuniverse generated by X is algebraic closure operator

This is taken from Universal Algebra Text book by Stan Burris. I have a question regarding the last conclusion as to how does the author conclude that Sg is an algebraic closure operator. How do ...
3
votes
1answer
52 views

Can a quasi-identity express that a function $f$ is surjective? And if not, can this be explained by duality?

Consider a first-order theory having a unary function symbol $f$. Then the following quasi-identity expresses that $f$ is injective. $$\forall xy : f(x)=f(y) \rightarrow x=y$$ Alternatively, we can ...
0
votes
1answer
79 views

What are the properties of this Poisson algebra?

I have the following (real) quantities (which are from a Classical Mechanics problem): $$A_1=\frac 1 4(x^2 +p_x^2-y^2-p_y^2 ) \quad A_2=\frac 1 2(x y +p_x p_y)$$ $$A_3=\frac 1 2(x p_y - y p_x )$$ ...
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0answers
17 views

A filterbase generating filter F

Show that a non empty subset $X$ of a filter $F$ in $B$ is a base for $F$ iff $X$ generates $F$ and for all $x,y$ $\in$ $X$ $\exists$ $z $ $\in$ $X$ such that $z$ $\leqq$ x $\wedge$ y.
0
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0answers
142 views

Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
1
vote
1answer
54 views

More Information about Magmas

I first learnt of magmas on Wikipedia and have been trying to read more on them just out of my own interest. Whenever I try to search them on Google, though, the search results are overwhelmed by the ...