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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Categories of relations over a fixed category $\mathcal{C}$

Let $\bf{Set}$ be the category of sets and functions. We have an associated category $\bf{Set}_\bf{Rel}$, whose objects are also sets but whose morphisms are relations, i.e. a morphism $X\xrightarrow{\...
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123 views

Lawvere algebraic theories as presentation-invariant

We can read in a lot of papers, included Lawvere's PhD thesis, that algebraic theories are "an invariant notion of which the usual formalism with operations and equations may be regarded as a ‘...
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155 views

Definition of $\Omega$-algebra

I'm studying universal algebra. I have this definiton: given a signature $\Omega$, an $\Omega$-algebra is comprised of a "carrier" for the algebra and an "interpretation" for every operation symbol. ...
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36 views

Lemma concerning compatibility of words (formed by a term algebra)

I need to prove the next lemma regarding compatibility of words in term algebras, that includes 3 parts: $u,v$ are compatible iff $u^ \smallfrown w_1= v^ \smallfrown w_2$. If $u_1u_2$ and $v_1v_2$ ...
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33 views

If $B$ is an algebra, $X \subseteq B$ and $A$ the smallest subset that extends $X$, then $A= \bigcup_{n < \omega} A_n$

I need to prove the next thing: Let $\textbf{B} = \langle B, (b_i)_{i \in I}, (g_j)_{j\in J} \rangle$. Let $X \subseteq B$ and let $A$ be the smallest subset of $B$ which extends $X$ and is closed ...
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52 views

Algebraic systems

Are there any books about algebraic systems without having Mal'cev book? Are there books in general about the varieties and quasi-varieties?
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156 views

The congruence class of a distributive lattice

I need to prove the next thing: For every non-empty ideal $I$ of a lattice $L$ consider the relation $\theta(I)$ defined by: $ \theta(I) = \{⟨a, b⟩ : ( \exists c \in I) a \vee c = b \vee c\}$ Prove ...
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113 views

Relating categorical properties of arrows

Diagram is from some book on categories shelf (ID?)- It's a good visual to rembember. What are some obvious extensions of categorical properties? Ie valid in every category, eg: strong-, extremal-, ...
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119 views

Is the existence of finite biproducts a strengthening of commutativity?

Let $T$ denote a monosorted Lawvere theory (call its distinguished object $G$) equipped with a distinguished constant $0 : G \leftarrow 1$ that is "idempotent" in the following sense: for all arrows $...
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65 views

Name for classes of algebras closed under products and quotients

A class of algebras closed under products, quotetiens and subalgebras is a variety. Is there a name for a class of algebras closed under products and quotients? Could you refer me to any theorems ...
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81 views

Rel is a concrete category over Sets, but how to concretize that?

The traditional denotation of a structured set object is something like $(1)\qquad(X,\mathcal S_1,\dots\mathcal S_n)$ for some "structures" $\mathcal S_1,\dots\mathcal S_n$ on a set X. The modern ...
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Common conditions on functions to be morphisms. [closed]

When coming in contact with the concept of morphism one may start to wonder what makes different structured objects of the same kind to be similar in a "morphical" way. At least I did. Below ...
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93 views

An example of a free algebra in SP(K)

The following theorem is found in the book "Universal Algebra, Fundamentals and Selected Topics" by Clifford Bergman (pp.98). Theorem 4.28. Let $U$ be free for $K$ over $X$. Then, $U/\lambda_k$ is ...
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67 views

When do (universal-algebraic) varieties have (enough) injectives?

Following Zhen Lin's suggestion in my previous question, I ask this question separately. Suppose $\mathcal{V}$ is a variety in the sense of universal algebra. considered as a category. When does $\...
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97 views

Are varieties cocomplete?

Consider a variety $\mathcal{V}$ in a sense of universal algebra, i.e. algebras of some fixed signatures described by a set of identities. Then $\mathcal{V}$ can be thought of as a category with ...
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128 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
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1answer
56 views

Prove the identity in Ring of Integers Modulo Prime

I have many study tasks, but I do not have any example. Therefore, I do not know, how to solve these tasks. For example, I need prove, that: $\{ b \in \mathbb{Z}_{p^n} \mid b^2 =1\} = \{-1, 1 \}$, ...
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806 views

Which texts do you recommend to study universal algebra and lattice theory?

As I'm planning to study some algebraic logic (a lot of!), I found that some knowledge of universal algebra, lattice theory and boolean algebras is a must. I wonder if you have any recommendation to ...
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85 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 ...
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2answers
80 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 \...
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83 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 $I(L)$. ...
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185 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 ...
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46 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 ...
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108 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 ...
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1answer
104 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 ...
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38 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 $P_S\leq SP$...
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203 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: ...
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1answer
113 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 ...
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1answer
61 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 ...
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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 ...
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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 ...
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1answer
187 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 ...
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155 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 ...
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1answer
64 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 $a\in ...
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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 ...
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68 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 ...
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45 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 ...
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105 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 = ...
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197 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 ...
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1answer
52 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 $|G|=|F|$. Does $...
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1answer
78 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 ...
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1answer
64 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 ...
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1answer
77 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, ...
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103 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 C(...
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86 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 ...
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1answer
37 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, ...
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1answer
60 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 $\mathrm{supp} : U(F(K)) \rightarrow \mathcal{P}_\mathrm{fin}(K).$ ...
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1answer
54 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) \...
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177 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 ...
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195 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 ...