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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397 views

What is a subdirect product?

I'm having trouble understanding what a subdirect product is. Say $G$ is a subdirect product of $H=\prod H_i$ - this means that the homomorphisms $f_i:G\to H_i$ are surjective, which can be ...
2
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1answer
33 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 : ...
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2answers
121 views

The kernel of the kernel.

From Wikipedia-Entry on Equivalence Relatin:Lattices The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The ...
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115 views

Does axiomatizability in zeroth-order logic have important consequences?

If a theory is equationally axiomatizable, this has important consequences (that are studied e.g. in universal algebra). However, many theories fail to be equationally axiomatizable - examples ...
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80 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) $f_{i}(a_0,\dots,a_{n-1}) \notin M$ ...
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108 views

Building factor semigroup w.r.t. $I_n = \{ w \in A^+ : |w| \ge n \}$.

I don't understand the following factor semigroup. Consider the pseudovariety $N$ of nilpotent semigroups. For any finite alphabet $A$, let $I_n = \{ w \in A^+ : |w| \ge n \}$. Then $A^+ / I_n$ is ...
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2answers
101 views

Categories without identities

What's the name of "categories without identities", i.e. of digraphs with just an associative binary operation on its "matching" arrows (disregarding identities)?
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2answers
270 views

Smallest Congruence Relation generated by a set

$\newcommand{\cl}{\operatorname{cl}}$ Let $R \subset S \times S$ be a binary relation, the smallest i) reflexive relation containing it is $$ \cl_\mathrm{ref} = R \cup \{ (x,x) : x \in S \} $$ ii) ...
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1answer
73 views

Proof that the lattice of fully invariant congruences is a sublattice of the lattice of all congruences

Let $\mathfrak U$ be an algebra (i.e. a set, called universe, together with several $n$-ary operations) in the sense of universal algebra. Denote by $\operatorname{Con} \mathfrak U$ the set of all ...
3
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3answers
339 views

New kind of identities?

I found a new kind of identities which are half logic and half algebraic while working on a proof of NP-completeness. They are like this: $$ \frac{a+mb}{n+m} < \frac{a}{n} \iff b < ...
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1answer
53 views

Why $F_{(\Omega,E)}(X)$ is a $\Omega$-structure?

I am reading the book "Notes on logic and set theory". It defines an $n$-ary equation, in an operational type $\Omega$, to be an expression $(s=t)$ where $s$ and $t$ are elements of $F_\Omega(X_n)$. ...
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73 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: $$R(x,y) \rightarrow R(f(x),f(y)) = R(x',y')$$ ...
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2answers
130 views

What is a free $\Omega$-structure generated by $X$

I am reading the book "Notes on logic and set-theory". It gives the following definition: Given a set $X$ of variables and operational type $\Omega$ ($X \cap > \Omega = \emptyset $), the set $ ...
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1answer
77 views

Isomorphic two structures but different type.

If I have structure $(S, \cdot)$, where $\cdot$ has type $(2)$, i.e., $\cdot : S \times S \rightarrow S$ and $(S', \circ)$, where $\circ$ has type $(3)$, i.e., $\cdot : S' \times S' \times S' ...
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1answer
51 views

Existence of arbitrarily large ordinal subgroups in a group structure on a regular cardinal [duplicate]

Suppose $\kappa$ is an uncountable regular cardinal, and $(\kappa, \cdot, ^{-1}, e$) is a group. Prove that that $C = \{\alpha < \kappa: \alpha\, \textrm{is a subgroup of}\, \kappa)$ is unbounded ...
6
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1answer
123 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 / ...
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4answers
300 views

Preserving structures

Category theory abstracts the notion of the preservation of structure by means of morphisms. Is there a description of what it means to preserve structure of different types of mathematical structures ...
6
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4answers
240 views

Any commutative associative operation can be extended to a function on nonempty finite sets

This is a fact we use very frequently in general mathematics when we write such notations as $1+2+3+4$: since we know that $+$ is commutative and associative, we can just "drop the parentheses" and ...
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2answers
216 views

In a slice category C/A of a category C over a given object A, What is the role of the identity morphism of A in C with respect to C/A

In a slice category $C/A$ of a category $C$ over a given object $A$, what is the role of the $C$ identity morphism, $A\to A$ ($1_A$), in $C/A$, particularly with respect to composition? I ...
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1answer
225 views

Property: closure under an binary operation

We have the following definition: Definition: let $*_A:A^2 \rightarrow A$ and $B \subseteq A $, with $B \neq \emptyset$, $B$ is closed under $*_A$ if $a *_A|_Bb \in B$ $\forall a,b \in B$. Property: ...
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1answer
113 views

Algebraic substructure and restriction of a function

I am reading the follow pdf: http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf in particular at pg 28 of pdf, and I think that, let $(A;f)$ an algebric structure and $B ...
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1answer
45 views

Do cancellative semigroups form a variety of algebras?

Sorry if this is a silly question. Define that a right-cancellative semigroup is a set $G$ together with an associative operation $*$ such that for all $a,b,x \in G$ it holds that $ax=bx \Rightarrow ...
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1answer
850 views

Modus Ponens: implication versus entailment

Would it be inconsistent to write Modus Ponens using only implication, not entailment? $(p \wedge (p \to q)) \to q$ The way I understand is that implication ($ \to$) is an operator that yields a new ...
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126 views

Represent the three element chain as a subdirect product of subdirectly irreducible lattices.

Represent the three element chain as a subdirect product of subdirectly irreducible lattices. I know the two element chain as either a Boolean algebra or a semilattice,is subdirectly irreducible.In ...
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1answer
138 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 ...
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0answers
21 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 ...
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1answer
143 views

Initial structures in the category of algebraic systems of the same type

From Handbook of Analysis and Its Foundations by Eric Schechter 9.21. Basic properties of subalgebras. We consider the category consisting of the algebraic systems of some type $(τ, \mathcal{J})$, ...
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4answers
713 views

why is a nullary operation a special element, usually 0 or 1?

Does a nullary operation mean an operation not taking any argument? Then why is a nullary operation a special element, usually 0 or 1, in an algebraic structure? Thanks!
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43 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 ...
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0answers
70 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 ...
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486 views

A structural proof that $ax=xa$ forms a monoid

During the discussion on this problem I found the following simple observation: If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid. This is trivial to prove by checking ...
3
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3answers
169 views

Free object is a coproduct: $F_{A\cup B}\cong F_A \coprod F_B$

Let $A,B$ be sets, and $A\sqcup B$ the disjoint union. Suppose that in a (concrete) category, the free objects $F_A,F_B,F_{A\sqcup B}$ exist, and that the coproduct $F_A \coprod F_B$ exists. How can I ...
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1answer
179 views

presentation of the direct sum of commutative rings / algebras

If $I,J$ are index sets, $R$ a commutative unital ring, $\mathfrak{a},\mathfrak{b}$ ideals of polynomial rings $R[x_i; i\!\in\!I]$, $R[y_j; j\!\in\!J]$, and $\langle\langle\ldots\rangle\rangle$ is the ...
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1answer
93 views

Describing all subdirectly irreducible mono-unary algebras.

(Wenzel). Describe all subdirectly irreducible mono-unary algebras. [In particular show that they are countable.] Thanks!
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1answer
94 views

Why $Z(A)$ is an equivalence relation on $A$?

For every algebra $A$, the center $Z(A)$ is a congruence on $A$. Why is $Z(A)$ an equivalence relation on $A$?
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1answer
142 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 ...
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2answers
72 views

Quotients of quotients in universal algebra

In universal algebra, when is the quotient of a quotient of an algebra $\mathcal{A} $, a quotient of $\mathcal{A} $?
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1answer
64 views

Proof for Tarski theorem in universal Algebra page 108

Given a variety V and a set of variables X, IrB(Idv(X)) is a convex set. I need a complete proof for this theorem. If anyone can help me it would be wonderful.
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1answer
50 views

Equational theories & their relation to fully invarient congruences on T(X)?

The equational theories of type F over X form an algebraic lattice which is isomorphic to the lattice of fully invarient congruences on T(X). I need the proof of above theorem which is in page 103 ...
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2answers
121 views

Why fully invariant congruence is an algebraic closure operator?

If we have an algebra $A$ of type $F$ then congruence of fully invariant is an algebraic closure structure operator on $A\cdot A$. Actually it's in Universal Algebra Sankappanavar page $100$ (Lemma ...
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1answer
42 views

My question is why the set of fully invariant congruence of a set A is closed under arbitrary intersection?

Why ($\operatorname{Con}_{FI}(A))$ is closed under arbitrary intersection?
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0answers
80 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 ...
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1answer
232 views

An exercise in infinite combinatorics from Burris and Sankappanavar

Exercise 6.7 in chapter IV of Burris and Sankappanavar's A Course in Universal Algebra starts as follows: Show that for $I$ countably infinite there is a subset $S$ of the set of functions from ...
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1answer
160 views

Unification of expressions involving sets

Let's let $\def\OP#1#2{\left\langle#1,#2\right\rangle}\OP xy$ represent the set $\{\{x\},\{x,y\}\}$ as is usual, per Kuratowski. Then: $$ \begin{eqnarray} \OP{\OP ab}c & = & \{\{\{\{a\}, ...
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1answer
70 views

Associativity, Jacobi, and self-action representations

About a year and a half ago, I was at a talk by Martin Hyland where he suggested that the Jacobi identity is to the associative law as the anticommutative law is to the commutative law. I think this ...
3
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1answer
173 views

Do filters on a Boolean algebra also make a Boolean algebra?

Let $\mathfrak{B}=(B,\bot,\top,\lnot,\wedge,\vee)$ be a boolean algebra. $B_F$ be the set of all filters on $\mathfrak B$. And for all filter $F$, $G$, $F \wedge_{B_F} G \colon= \mathbf C(F \cup G)$ ...
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2answers
196 views

Stone duality for ideals and filters (exercise)

In A Course in Universal Algebra (Burris, Sankapannavar), the exercise 4.4.7-8, p.158, says: Let $A$ be a Boolean algebra. Denote $A^\ast:=\{\text{ultrafilters of }A\}$, and give $A^\ast$ the ...
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4answers
442 views

Are isomorphic structures really indistinguishable?

I always believed that in two isomorphic structures what you could tell for the one you would tell for the other... is this true? I mean, I've heard about structures that are isomorphic but different ...
2
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1answer
150 views

correspondence for universal subalgebras of $U/\vartheta$

Let $U$ be a universal algebra of type $T$, and denote $\mathrm{Con}(U)\!=\!\{\text{congruence relations on }U\}$ and $\mathrm{Sub}(U)\!=\!\{\text{subalgebras of }U\}$. Let "$\leq$" mean "subalgebra". ...
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263 views

What is the name of the structure Z4 under subtraction?

If we consider $\mathbb{Z_4}$ under addition, then it forms a cyclic group of order 4. However if we change the binary operation to subtraction on $\mathbb{Z_4}$, we get a different structure $J$ with ...