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).

learn more… | top users | synonyms

5
votes
1answer
83 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 ...
5
votes
1answer
122 views

How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...
3
votes
2answers
134 views

For what categories do category algebras exist?

The monoid algebra $R[M]$, for a commutative ring $R$ and a monoid $M$, can be described as the free $R$-algebra on $M$. We think of $R[M]$ as the set of finite formal sums of elements of $M$ with ...
3
votes
2answers
136 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
0
votes
1answer
59 views

Basic question about the definition of a variety (in universal algebra)

According to wikipedia, A variety is a class of algebraic structures of the same signature that is closed under the taking of homomorphic images, subalgebras and (direct) products. Isn't ...
5
votes
1answer
54 views

Which sentences survive the passage from $X$ to the set of all functions $I \rightarrow X$?

Suppose $X$ is a mathematical structure with a single underlying set which we will also denote $X$, equipped with some functions and relations. Letting $I$ denote an arbitrary non-empty set, we see ...
0
votes
1answer
89 views

Relation between quotients and subalgebras

If I have two algebras $A,B$, and one is the quotient of the other, i.e. there exists a surjective morphism $\phi : A \to B$. Then is $B$ isomorphic to some subalgebra of $A$? I think so, because I ...
3
votes
1answer
117 views

Completing a Partially Defined Associative Binary Operation

This is more like a question about terminology. I would like to hear some recommendations of books that discuss algebraic structures with one partially defined associative binary operation, and the ...
3
votes
0answers
90 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 ...
1
vote
0answers
81 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 $A$ ...
2
votes
2answers
198 views

Isomorphism of algebras $\mathbb{Q}(\cdot , +, 1)$ and $\mathbb{Z}(\cdot , +, 1)$

I have these two algebras and I need to know if they are isomorphic: $\mathbb{Q}(\cdot , +, 1)$ and $\mathbb{Z}(\cdot , +, 1)$ Are there some general tricks how to deal with this type of tasks?
1
vote
2answers
63 views

Notation for “incommensurate” elements?

Say that $x\wedge y\not=x,y$, that is neither $x\leq y$ nor $y\leq x$. Is there a way to denote this? I've been saying $x<>y$ but that's completely made up.
1
vote
1answer
512 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
votes
1answer
37 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 : ...
1
vote
2answers
123 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 ...
2
votes
1answer
129 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 ...
2
votes
0answers
95 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
votes
2answers
111 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 ...
3
votes
2answers
120 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)?
2
votes
2answers
340 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) ...
1
vote
1answer
86 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
votes
3answers
347 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 < ...
1
vote
1answer
56 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 $F_\Omega(X_n)$. ...
0
votes
0answers
92 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: $$R(x,y) \rightarrow R(f(x),f(y)) = R(x',y')$$ ...
1
vote
2answers
133 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 $ ...
0
votes
1answer
78 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' ...
1
vote
1answer
61 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
votes
1answer
127 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
4answers
409 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
votes
4answers
271 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 ...
7
votes
2answers
348 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 ...
1
vote
1answer
320 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: ...
1
vote
1answer
122 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 ...
1
vote
1answer
50 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 ...
6
votes
1answer
1k 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 ...
0
votes
1answer
155 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 ...
2
votes
1answer
171 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 ...
0
votes
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 ...
1
vote
1answer
168 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 $(τ, \mathcal{J})$, ...
4
votes
4answers
1k 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!
0
votes
0answers
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 ...
1
vote
0answers
73 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 ...
19
votes
2answers
499 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
votes
3answers
181 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 ...
-2
votes
1answer
190 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 ...
0
votes
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 $A$?
1
vote
1answer
159 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 ...
1
vote
2answers
77 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} $?
0
votes
1answer
69 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.
0
votes
1answer
52 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 ...