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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13 views

Show that the unary algebra$<M,.,1,(f_{a})_{a}\in A>$ satisfies $f_{a_{1}…a-{n}}\approx f_{b_{1}…b-{n}}\leftrightarrow?$ [on hold]

answer:$<M,.,1>$ is a algebra with binary and nullary operation satysfying $G_{1}$,$G_{2}$ $G_{1}$:$x.(y.z)=(x.y).z$ $G_{2}$:$x.1=1.x=x$ how to cntinue?
4
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1answer
55 views

Is this generalization of Boolean algebras a variety?

I'm interested in the class of structures $\langle S,\top,\neg,\wedge\rangle$ defined by the axioms: $p \wedge q=q \wedge p$ $p \wedge (q \wedge r) = (p\wedge q)\wedge r$ $p = p \wedge p$ $p = ...
2
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0answers
17 views

Clone of operations acting bicentrally

Let $(A,F)$ be an algebra, let $F^{\star}$ be the centralizer of $F$ and $F^{\star\star}:=(F^{\star})^{\star}$ the bi-centralizer. Let $[F]$ denote the clone of operations generated by $F$. We say ...
0
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0answers
20 views

how to find $X\cup E(X)\cup E^{2}(X) \cup …\subset Sg(x)$?

qUESTION:$Sg(x)=X\cup E(X)\cup E^{2}(X) \cup ...$ answer: $Sg(x)=\bigcap\{B:X\subset B and Bis Subunivers Of A\}$ $Sg(x)$ is subuniverse generated by $X$ and if $f$ is a fundamental $n$-ary ...
1
vote
3answers
37 views

Constructing the groupification of a semigroup (Vakil 1.5.G)?

In the first chapter of Ravi Vakil's Algebraic Geometry notes, he suggests a construction of the groupification $H(S)$ of an abelian semigroup $S$ by considering $S\times S/\sim$ where $(a,b)\sim ...
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0answers
9 views

Closure systems [duplicate]

Let A be any set. A system $\mathscr{C}$ of subsets of A is said to be a closure system if $\mathscr{C}$ is closed under intersections, i.e. $$\textrm{for any subsystem ...
7
votes
1answer
74 views

Distributor? Distributive analog of commutator and associator?

Motivation: "the commutator gives an indication of the extent to which a certain binary operation fails to be commutative" (http://en.wikipedia.org/wiki/Commutator). For example (courtesy of ...
0
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1answer
26 views

why $(\theta _{1}\circ \theta _{2})^{\vee }\subset(\theta _{2}\circ \theta _{1})^{\vee } $

my questiones at this theorem: i coud not undrestand $a\Rightarrow b$ and why $(\theta _{1}\circ \theta _{2})^{\vee }\subset(\theta _{2}\circ \theta _{1})^{\vee } $ please guide me?
0
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0answers
34 views

Show Sg$(X) = X\cup E(X) \cup E^{2}(X)\cup …$

Given an algebra $A$ define, for every $X\subset A$, $$\text{Sg}(X)=\bigcap\{B\mid X\subset B \text{ and } B \text{ is a subuniverse of } A\},$$ $$E(X)=X\cup \{f(a_1,\ldots,a_n)\mid f \text{ is ...
0
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0answers
18 views

how to find N(G) is a modular lattice?

answer: $N_{1}\leq N_{2}$ so $N_{1}\vee(N_{2}\wedge N_{3})=(N_{1}\vee N_{2})\wedge (N_{1}\vee N_{3})=N_{2}\wedge (N_{1}\vee N_{3})$ how to continue?
0
votes
0answers
25 views

why $P$ is not a sublattice of the lattice $\{a,b,c,d,e\}$

I could not understand why $P$ is not a sublattice of the lattice $\{a,b,c,d,e\}$ how to find?
0
votes
1answer
37 views

how to find $\alpha(a\wedge b)=\alpha(a)\wedge\alpha(b)$

for $\alpha(a\wedge b)=\alpha(a)\wedge\alpha(b)$ how to do? answer: $\alpha $ and $ \alpha^{-1}$ are order-preserving and $a \leqslant b$ and $a=a\wedge b$ so $\alpha(a) = \alpha(a\wedge b)$ so ...
0
votes
1answer
41 views

show that N(G) is a lattice?

if $G$ is a group let N(G) be the set of normal subgroups of $G$ define $\wedge$ and $\vee$ on$ N(G)$ by $N_{1}\wedge N_{2}=N_{1}\cap N_{2}$ and $N_{1}\vee N_{2}=N_{1} N_{2}=\{n_{1}n_{2}:n_{1}\in ...
2
votes
1answer
70 views

Dual atoms of the lattice of varieties

I'm reading Jaroslav Ježek's "Universal algebra". There is a Theorem. For a signature containing at least one symbol of positive arity, the lattice of varieties of that signature has no coatoms. ...
1
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2answers
65 views

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 ...
0
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0answers
39 views

If $G$ is a group, show that the lattice $N(G)$ of normal subgroups of $G$ is a modular lattice

If $G$ is a group, show that the lattice $N(G)$ of normal subgroups of $G$ is a modular lattice. Does the same property holds for the lattice $S(G)$ of all subgroups? Describe $N(Z_{2}\times Z_{2})$.
1
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1answer
83 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 ...
1
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0answers
36 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. ...
0
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0answers
21 views

Quoting a quasi-variety by a set of quasi-equations

Given a variety $V$, an algebra $A \in V$ and a set of equations $E$ one may form, in the standard way, the quotient algebra $A/E$. The algebra $A/E$ has the property that is satisfies all the ...
0
votes
1answer
33 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 $v_1v_2$ ...
2
votes
1answer
28 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 ...
0
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0answers
45 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?
0
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1answer
46 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 ...
2
votes
2answers
86 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-, ...
5
votes
1answer
67 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 ...
2
votes
0answers
56 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 ...
0
votes
1answer
61 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 ...
0
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1answer
82 views

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 ...
0
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0answers
82 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 ...
3
votes
0answers
55 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 ...
3
votes
2answers
76 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 ...
1
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0answers
106 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 ...
1
vote
1answer
41 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: $\{ b \in \mathbb{Z}_{p^n} \mid b^2 =1\} = \{-1, 1 \}$, ...
3
votes
4answers
259 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 ...
3
votes
1answer
55 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
70 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
70 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 $I(L)$. ...
3
votes
2answers
137 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
42 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 ...
1
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1answer
78 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
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1answer
88 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
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1answer
33 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 ...
4
votes
1answer
114 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
100 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
40 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
91 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
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0answers
92 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
163 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
108 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
51 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 ...