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6
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1answer
116 views

What is a simple axiomatisation of groups using division?

I recall from an old exercise I did as an undergrad that groups can be axiomatised using division rather than multiplication: A group is a non-empty set equipped with a binary division operator / ...
4
votes
1answer
80 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 ...
4
votes
1answer
77 views

The “closed” subspaces of topological algebraic structures

Every set-theoretic model of an algebraic theory gives rise to notion of (algebraically) "closed subset" in a canonical fashion; namely, the closed subsets are those that cannot be escaped via the ...
2
votes
1answer
31 views

Local smallness of Lawvere theories

Reading this blog post, I'm trying to care about foundational matters. To summarize the first part of the article, living in a univers $\mathcal V$ of sets, one defines a Lawvere theory as follow : ...
2
votes
1answer
121 views

Every group and ring is congruence-permutable , but not necessarily congruence-distributive

The problem is: Show that every group and ring is congruence-permutable , but not necessarily congruence-distributive. I know that in group every normal sub group has permutable property and in ...
1
vote
1answer
69 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
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1answer
45 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.
1
vote
1answer
124 views

homomorphisms and congruence relations

Do compositions of homomorphisms in universal algebra correspond to joins of congruence relations? That is- is the congruence relation $g \circ f(a ) = g \circ f( b) \Leftrightarrow a \sim b $ the ...
0
votes
1answer
82 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 ...
-1
votes
1answer
84 views

Describing all subdirectly irreducible mono-unary algebras.

(Wenzel). Describe all subdirectly irreducible mono-unary algebras. [In particular show that they are countable.] Thanks!
5
votes
0answers
46 views

Is there a name for the algebra of substructures?

Let $X$ denote an entropic algebra (see here), which just means that all the operations of $X$ are homomorphisms $X^n \rightarrow X.$ Abelian groups are the classic example. Then for any operation of ...
4
votes
0answers
79 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 ...
3
votes
0answers
51 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
0answers
86 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 ...
3
votes
0answers
46 views

Seeking information about (category-theoretic) varieties, quasivarieties, and universal Horn classes.

I'm looking for a list of basic facts regarding (category-theoretic) varieties, quasivarieties, and universal Horn categories, as well as information about which forgetful functors preserve what. In ...
3
votes
0answers
87 views

What big families of theories/structures are there?

To gain a big picture of (pre-categorical) mathematics, is it correct to divide mathematical theories resp. structures in two big families? universal algebra: classes of objects with arbitrary ...
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 ...
2
votes
0answers
93 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 ...
2
votes
0answers
70 views

Which algebraic identities survive the journey to the powerset?

Given an algebraic structure $A$ (call its underlying set $U$) we can obtain a new algebraic structure $B$ with underlying set $V=\mathcal{P}(U)$ in the obvious way. In particular, if $f : U^n ...
2
votes
0answers
62 views

A subalgebra of $P(M, \Delta)$ is a Peano Algebra

Let $M$ be any set and let $\Delta=(n_i)_{i \in I}$ be an algebraic type. Let $P=P(M, \Delta)$ be the algebra such that the generalized Peano Axioms hold: (P1) $f_{i}(a_0,\dots,a_{n-1}) \notin M$ ...
1
vote
0answers
93 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
0answers
38 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
vote
0answers
36 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
0answers
28 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 ...
1
vote
0answers
154 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
0answers
53 views

What is the $K$-free algebra for the class of implication algebras, over a finite set

I suppose the title is pretty self explanatory. I have been struggling with the concepts of $K$-free algebras, where $K$ is some class of same-type algebras, over some set $X$. So, in trying to ...
1
vote
0answers
17 views

Does the notion of “commutative algebraic theory” generalize to many-sorted signatures?

I think that the notion of commutative algebraic theory only makes sense for unisorted signatures, and cannot sensibly be generalized to the case of more than one sort. Does anyone know of a ...
1
vote
0answers
71 views

Congruence lattice of a partial algebra is algebraic

A partial operation on a nonempty set $A$ is a map $f:\mathrm{dom}(f,A)\to A$ where $\mathrm{dom}(f,A)\subseteq A^n$ for some $n\in\mathbb{N}$. A partial algebra is an ordered pair $(A,P)$ where $A$ ...
1
vote
0answers
66 views

Which “conditions” generate subalgebras?

While looking at this question I suddenly wondered about a more general question. Think of your favorite class of algebra (groups or rings, say), and forgive me for vocabulary mistakes while I try to ...
1
vote
0answers
76 views

Another way of saying that algebraic objects are isomorphic

From a universal algebraic perspective, let's say we have two isomorphic groups. Then can I speak of their isomorphic nature by saying the binary operations of multiplication of the two groups are ...
1
vote
0answers
84 views

direct product of different algebras?

Is it possible to define a "direct product" of two algebras with different signatures? For example, boolean lattice $\otimes$ monoid? Perhaps we need to take some quotient to make sense of it (e.g Q ...
1
vote
0answers
131 views

amalgam of structures

Trying to refine my question here. This is a response to the questions here: Homomorphisms between structures My objective is to take a set of $S-$structures and form an amalgam object out of that ...
0
votes
0answers
53 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 ...
0
votes
0answers
81 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 ...
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 $T$. ...
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 ...
0
votes
0answers
27 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 $z \rightarrow \alpha z + \beta$ ...
0
votes
0answers
48 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 ...
0
votes
0answers
60 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
0answers
20 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
votes
0answers
58 views

Medium-strong (graph) homomorphisms

Weak (graph) homomorphisms are mappings $f: V(G) \rightarrow V(G')$ such that the images of connected nodes $x,y$ (in the source graph) are connected: $$R(x,y) \rightarrow R(f(x),f(y)) = R(x',y')$$ ...
0
votes
0answers
19 views

verify the claim that consequences of balanced identities are again balanced.

verify the claim that consequences of balanced identities are again balanced. An identity is p≈q balanced if each variable occurs the same number of times in p as in q.if ∑ is balanced set of ...
0
votes
0answers
39 views

Question about a property of lattice-morphism

I would like to know if there is a name for the class of commutative (i.e., $\phi(x,y)=\phi(y,x)$) lattice-morphisms $\phi : L_1\times L_{1} \rightarrow L_2$ with the following property: $\phi(x ...