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6
votes
1answer
110 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
73 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
29 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
96 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
104 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
71 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
37 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 ...
3
votes
0answers
44 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
83 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
20 views

Congruence lattice of a semilattice is meet-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 ...
2
votes
0answers
57 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
58 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
19 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
0answers
24 views

Variety satisfying an identity.

V is a variety of commutative semi group 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
0answers
29 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 ...
1
vote
0answers
17 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
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
63 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
55 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
71 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
82 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
109 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
10 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 ...
0
votes
0answers
20 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
37 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
53 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
31 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
0answers
12 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 ...
0
votes
0answers
15 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
137 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 ...
0
votes
0answers
38 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 ...
0
votes
0answers
52 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
18 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
32 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 ...
-1
votes
0answers
37 views

Power set as a spectral space

Let $X$ be an infinite set. Is there way to show that $2^{X}$ is a spectral space ?. Thanks.