The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
1answer
42 views

A conjecture in equational logic

In an algebra with a single binary operation g, is there a single equational identity that generates the same set of identities as the set {g(x,y)=g(y,x) , g(g(x,y),z)=g(x,g(y,z))}? My conjecture is ...
3
votes
2answers
56 views

Are $e : 1 \rightarrow X$ and $\mathrm{id}_X : X \rightarrow X$ the only operations of $\mathsf{Grp}$ that are homomorphisms?

Let $\mathsf{Grp}$ denote the Lawvere theory of groups. (For concreteness, let us say that $\mathsf{Grp}$ is presented by the generators $c : X \times X \rightarrow X$ $e : 1 \rightarrow X$ $i : X ...
1
vote
1answer
51 views

Showing finite lattices are isomorphic to their sets of ideals.

I'm working out a problem from Gratzer '71. We are to show that any finite lattice $L$ $\simeq$ $I(L)$. My attempt was as follow: prove the contrapositive. So assume $L$ is not isomorphic to $I(L)$. ...
1
vote
2answers
87 views

Algebraic Structures: Does Order Matter?

(I want to link to similar question with a very good answer: Question about Algebraic structure?) An algebraic structure is an ordered tuple of sets. One of these is called the underlying set, and ...
1
vote
0answers
37 views

Is there accepted terminology for algebraic structures whose every subalgebra is free?

Is there accepted terminology for algebraic structures whose every subalgebra is free? Examples: Any free group Any vector space More generally, any free module over a PID. In fact, this ...
2
votes
0answers
32 views

Does the phrase '$S$ is independent' have an accepted meaning in universal algebra?

Let: $T$ denote an algebraic theory $F$ denote the free functor $X$ denote a $T$-algebra. $\mathrm{cl}_X : \mathcal{P}(X) \rightarrow \mathcal{P}(X)$ denote the function such that for all $S ...
-1
votes
0answers
47 views

An ideal of a Boolean algebra

Theorem. Every ideal $I$ of a Boolean algebra $A$ determines a congruence $\theta$ of the Boolean algebra $A$ as follows: $a\theta b \Leftrightarrow \bar{a}b + a\bar{b} \in I$. Every congruence can be ...
1
vote
1answer
66 views

Existence of countably generated free algebra in the universal class

I'm trying to solve the following exercise (from Smirnov's "Varieties of algebras"): Problem: Let $K$ be the universal class of $\Omega$-algebras, i.e. $K = Mod(\Sigma)$, where $\Sigma$ is the ...
0
votes
1answer
76 views

The concept of K-free algebras

This is Definition 10.9 of the book "A Course in Universal Algebra" by Burris and Sankappanavar (page 73, Millennium Edition). http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf ...
2
votes
1answer
24 views

A Birkhoff theorem on K-free algebras

Let $F_K(\overline{X})$ be the $K$-free algebra over $\overline{X}$. I want to prove that $F_K(\overline{X})\in ISP(K)$. I have already proved that $F_K(\overline{X})\in IP_SIS(K)$. Since $P_S\leq ...
3
votes
1answer
70 views

Birkhoff's completeness theorem

I have two simple questions. A) Does Birkhoff's completeness theorem follow directly from Gödel's completeness theorem? B) Is Birkhoff's completeness theorem constructive in the following sense: ...
0
votes
1answer
93 views

K-free algebra over $\overline{X}$

This is Definition 10.9 of the book "A Course in Universal Algebra" by Burris and Sankappanavar (page 73, Millennium Edition). http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf ...
1
vote
1answer
30 views

What does a lattice of the direct power of the two-element chain look like?

In universal algebra, it is known that every finite Boolean lattice is isomorphic to a direct power of the two-element chain. I am having hard time figuring out what a lattice of the direct power of ...
3
votes
0answers
82 views

Prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$ where $\{I_j \ | \ j \in J\}$ is a partition of I.

My problem is following: prove that $\Pi_{i \in I} \mathbf{A}_i \cong \Pi_{j \in J} (\Pi_{i \in I_j} \mathbf{A}_i)$, where $\langle \mathbf{A}_i \ | \ i \in I \rangle$ is an indexed set of similar ...
0
votes
0answers
78 views

A Course in Universal Algebra (Millennium edition), page 74

The line before Theorem 10.12 says that "In general $F_K(\overline{X})$ is not isomorphic to a member of K (for example, let K={L} where L is a two-element lattice, then $F_K(\bar{x}, \bar{y}) \notin ...
5
votes
1answer
142 views

What is the most expressive logic such that presentations of algebraic structures “work”?

I feel like this is one of the best questions I've asked in a while. Hope you enjoy it. In my opinion, one of the most important ideas in modern algebra is the idea that we can present algebraic ...
4
votes
1answer
68 views

Reference for understanding coalgebra

I am trying to read this paper, but I have no knowledge of coalgebra and have just started to learn Category Theory so I am struggling to understand it. Are there any references that can explain ...
1
vote
1answer
46 views

Irreducible elements in a Lattice.

Let $L$ be a lattice. We say that $a\in L$ is irreducible if for every $b,c\in L$ such that $a=b\vee c$ we can conclude that $a=b$ or $a=c$. If $L$ is a finite lattice prove that every element ...
4
votes
0answers
78 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
2
votes
1answer
47 views

Reference for $F$-algebras and induction?

I've been learning about $F$-coalgebras and coinduction from this fantastic paper, which has really helped me get a feel with its many examples. I'm starting to struggle with reconciling the ...
0
votes
1answer
29 views

Reducing Several Identities to One Identity

One class of algebraic structures that are typically studied are those given by a set $X$ and a set of $n$-ary operations defined on $X$ for each $n\in \mathbb{N}$. Perhaps most studied are those ...
3
votes
1answer
67 views

Subalgebras of free algebras

I am trying to prove the following statement: Problem: There is no free groups in the universal class $\mathcal{A}$ of all abelian groups satisfying $\forall x (x + x = 0) \vee \forall x (x + x + x = ...
6
votes
1answer
115 views

Using the compactness theorem to disprove axiomatizability

Another model-theoretic exercise from Smirnov's book. Problem: Construct infinite family of varieties such that their union is not axiomatizable. My solution: Denote by $\mathcal{A}_n$ the variety ...
4
votes
1answer
43 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 $|G|=|F|$. ...
3
votes
1answer
64 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 ...
3
votes
1answer
43 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
64 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
31 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
52 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
29 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
49 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
43 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
150 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
143 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
45 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
108 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
148 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
55 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
90 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 ...
3
votes
1answer
112 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 ...
2
votes
1answer
84 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 ...
0
votes
0answers
24 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
169 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
105 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
83 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
30 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
votes
2answers
139 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
35 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 ...