-1
votes
0answers
12 views

simple exercise in Cylindric algebra

I am trying to gain a better understanding of cylindric algebra, so I made up this example. Given a general rule that someone's father's father is his/her grandfather: $\forall_X ~ \forall_Y ~ ...
2
votes
1answer
36 views

Is there a phrase to describe those objects of $\mathbf{C}$ that can be expressed as quotients of the algebra freely generated by $X$?

Let $\mathbf{C}$ denote the category of models of an algebraic theory in $\mathbf{Set}.$ Now suppose $X$ is an object of $\mathbf{Set}$. Is there a traditional phrase used to describe those objects of ...
1
vote
2answers
39 views

Abelian/Isomorphic logic statement

I don't understand this logic statement. I don't think the context helps at all but I thought i'd include it anyway. $G$ is abelian and $H$ is not abelian, then $G\ncong H$, is the same as: $G$ ...
2
votes
2answers
111 views

Proof of Sylow's theorem.

I read this proof of Sylow's theorem in Rotman's "An introduction to the Theory of Groups" and I don't understand what is the argument in the second paragraph (the one in the green box) for. Isn't ...
1
vote
1answer
75 views

Definition of a group written in formal logic

Is the formal logic in these axioms correct? A group is a nonempty set $G$ with a binary operation $*$ that satisfies the following axioms: $(a * b) \in G \; \forall \;a,b \in G$ $a * (b * c) = (a ...
1
vote
0answers
28 views

Boolean algebras without the least element

I read somewhere that A.Tarski introduced Boolean algebras without the least element, but could not find a free article about them. If anybody knows the axioms of such an algebra, can you please ...
3
votes
1answer
41 views

Some questions on interdependence of some properties of abstract magma

Does there exist a magma $(S,\cdot)$ such that for every $y\in S, \exists y'\in S$ such that $x\cdot(y\cdot y')=x, \forall x,y\in S$, but there exist $x_1, x_2, x_3\in S$ such that ...
1
vote
1answer
29 views

Can covering be done on two elements?

The covering rule is: $$B \bullet (B+C) = B$$ and $$B+(B \bullet C)=B$$ So does it follow from this rule that: $$B \bullet A \bullet \bar{C} + B \bullet D \bullet\bar{F} = B \bullet ...
11
votes
2answers
228 views

Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, ...
2
votes
2answers
62 views

Is Dihedral group just for small order?

I'm sorry that the title is not really specifying a question, but i cannot think of a single sentence describing this question. Dihedral group is defined to be rotations and reflections of $n$-agon ...
1
vote
1answer
61 views

Do the circle groups have any interesting stand-alone descriptions?

By the circle groups, I mean firstly the circle group $\mathbb{T} \subseteq \mathbb{C}$ of all complex numbers having modulus $1$, and secondly the commutative group $\mathbb{S} = \mathbb{T} \cap ...
0
votes
0answers
27 views

Resources for Polyadic and/or Cylindric Algebra

I'm looking to learn a little bit about polyadic and cylindric algebras, as part of an investigation into algebraic approaches to logic. The only "text" that I can find for polyadic algebra is ...
0
votes
1answer
38 views

Two questions about the lattice derived from 0th-order formulas

It's not clear to me if the definitions I've been given are common. Therefore I will give a brief overview of the constructions I'll need to talk about the objects I want to. Prerequisite: Given ...
1
vote
2answers
49 views

Do “equational theories” include sequents?

In equational logic, which of the following best describes the term "equational theory"? A collection of identities. A collection of quasi-identities, by which I mean sequents of the form ...
2
votes
1answer
58 views

Algebraic signatures as quivers; is there somewhere I can learn more about these definitions?

In my opinion, a cool definition of "algebraic signature" is as follows: An algebraic signature on the sort symbols $\mathcal{X} = \{X_0,...,X_{n-1}\}$ is precisely a quiver whose underlying set ...
6
votes
1answer
96 views

Have mathematical structures equipped with “generalized relations” been considered in a systematic way?

A binary relation on $X$ is basically just a function $X^2 \rightarrow \mathbb{B}$, where $\mathbb{B}$ is the prototypical Boolean algebra $\{0,1\}.$ We can generalize by replacing $\mathbb{B}$ with a ...
2
votes
0answers
54 views

Formulas in a Field and in a Field Extension.

Let $\mathbb F$ be a field and let $a, b, c, d$ be fixed elements in the field $\mathbb F$. Consider the formulas 1) $\exists\;x\;\;:\;\;x^2=-1.$ 2) $\exists\;x\;\;:\;\;(xa=c\land xb=d).$ Formula ...
1
vote
1answer
64 views

Lindenbaum's lemma and algebraic closures

Lindenbaum's lemma in predicate logic states that every consistent, first order theory $K$ has an extension to a consistent, complete extension. This sounds very similar to the theorem that every ...
1
vote
1answer
74 views

How many distinct functions for a set containing four elements? [closed]

How many distinct unary and binary functions can be defined on a set containing four elements? Edit: How many distinct unary and binary operations can be defined on a set containing four elements?
5
votes
1answer
149 views

Can logic be significantly geometrised?

I've already asked this question on philosophy.stackexchange, I'm hoping for a different answer here: Descarte has been lauded for putting together geometry and algebra, and his achievement allowed ...
2
votes
0answers
56 views

Difference between defining a constant and beginning with it in a structure

For example, let's suppose that I have my structure $\langle\mathbb{R},+\rangle$ and that $\exists!x\forall a\in \mathbb{R}(a+x=x+a=a)$ as an axiom. In this case $0:=x$. But what if I consider the ...
3
votes
0answers
61 views

Are there some kind of “multialgebras” with terms or equations, where an operation can result with different values in different places?

Many-valued (multivalent, polivalent) operations are studied in multialgebras. Applied to a certain value of its argument, a many-valued operation o(x) can result in different values. But in ...
1
vote
1answer
56 views

Where can I learn more about these two functions obtained from IFF and XOR?

Given a set $X$, write $\mathrm{heaps}(X)$ for the set of all finite heaps (or 'multisets', if you prefer) on $X$. Under this definition, it is well-known that if a binary operation $*$ on a set $X$ ...
2
votes
1answer
84 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
2answers
212 views

What are authoritative publications regarding foundational mathematics?

I have a computer science background. In our world, there usually is an organization publishing standard documents for certain areas (e.g. W3C has Web standards, IETF publishes Internet-related ...
4
votes
1answer
215 views

How far can we get without a foundation, using just first-order logic?

I think its interesting to ask how far we can get without committing to any particular foundations, using just first-order logic. For instance, we can prove theorems in this way about partially ...
2
votes
1answer
93 views

Gap in Halmos & Givant's “Logic as Algebra”: undefined $\models$?

I have been hugely enjoying Logic as Algebra by Halmos and Givant (1998, isbn: 0-88385-327-2)1, largely because I appreciate the authors' careful attention to the ...
-2
votes
1answer
86 views

Logic and its connection [closed]

My main interests are Algebra,Complex (fourier/laplace) analysis and Differential equations. In the future i plan to know a lot of abstract algebra/linear algebra/universal algebra/functional ...
4
votes
1answer
67 views

Class of finite groups a Fraïssé Class? [duplicate]

Is the class of finite groups a Fraïssé class? Calling this class $K$, does $K$ satisfy the following: Joint embedding property Amalgamation property Hereditary property: if $G \in K$ and $H \le G$, ...
5
votes
2answers
172 views

What are the main relationships between exclusive OR / logical biconditional?

Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain. Its well known that both exclusive OR and logical biconditional make $\mathbb{B}$ into an Abelian group (in the former case the identity is $0$, ...
3
votes
0answers
137 views

meaning of ``partial converse''

In the definition of a commutative ring $(R,+,\times)$, one of the postulates given is that of uniqueness, i.e. that $$ a=a', b=b'\implies a+b=a'+b', ab=a' b'.$$ The text states that for the system ...
1
vote
1answer
126 views

Name of “inverse distribution” axiom?

We may speak of a function distributing over another, e.g., $$ f(a,g(b,c)) = g(f(a,b),f(a,c)) \qquad a \cdot (b + c) = a \cdot b + a\cdot c$$ Some logical operators have similar distribution rules, ...
3
votes
2answers
95 views

Non-Archimedan Groups

I'm trying to think of an explicit example of a non-Archimedian group. The definition of Archimiedean is s.t. if for all $x$ and $y$, there is some $m$ a Natural number s.t. $mx = \underbrace {x + x ...
1
vote
1answer
74 views

Coding a sequence into a natural number by map $f$ with $f(k_1, .., k_n) + f(k_{n+1}, .., k_{2n}) = f(k_1 + k_{n+1}, .., k_n + k_{2n})$

Has anyone discovered a way of coding a sequence of natural numbers into a natural number by map $f$ that has the following property $f(k_1, .., k_n) + f(k_{n+1}, .., k_{2n}) = f(k_1 + k_{n+1}, .., ...
4
votes
2answers
64 views

Can we prove that an extension to a structure is consistent if the original structure is?

As I understand it, there is no way to prove that $\mathbb{N}$, as modeled by P.A., is consistent - meaning it may be possible to demonstrate eg. $5 = 3$. Therefore it is presumably also impossible to ...
5
votes
3answers
212 views

How can we tell if a set of axioms uniquely determines an algebraic structure?

Up to isomorphism. For instance, the group axioms are verified by an infinite number of non-isomorphic algebraic structures. But the Peano axioms, I think (my proof may lack some formality due to my ...
1
vote
2answers
182 views

How can we know arithmetical axioms are consistent?

If we assume both distributivity and the opposite of the law of signs (ie, that $-1\times-1 = -1$) for the relative integers, then we can derive that two different numbers are actually equal. ...
3
votes
2answers
134 views

Category-Theory and the modelling of $n$-ary functions (especially the $0$-ary functions)

Hi have a questioning regarding the modelling of $n$-ary functions and constants. First in the category of sets I know that the empty set $\emptyset$ is initial because for every other set $X$ there ...
2
votes
1answer
105 views

Understanding the model-theoretic proof of Hilbert's Nullstellensatz

The proof I am talking about goes like this: Given $k$ algebraically closed and $(f_1,..,f_k)=I\neq (1)$ an ideal in $A=k[x_1,..,x_n]$, let $m$ be a maximal ideal with $I\subseteq m$ and observe that ...
1
vote
1answer
145 views

Is $\mathbb N$ definable in $\mathbb C$?

$\mathbb C$ is an algebraic closed field with characteristic $0$, hence $Th(\mathbb C)$ is a recursive satisfiable complete theory, thus recursive axiomatizable. So if $\mathbb N$ is definable in ...
2
votes
2answers
49 views

Definable orders

Let $(K, <)$ be an order field, can I define the order "<" in $K$ ? I know that $K \models 0<a \;$ if and only if there is $b$ in the real closure of $K$ such that $b*b = a$. Can I ...
0
votes
2answers
33 views

A logic statement. “or” in Abstract algebra - groups

Let H be the subset of $M_2(\mathbb{R})$ consisting of all matrices of the form $H^* = \left \{ \begin{pmatrix} a &-b \\ b&a \end{pmatrix} : a,b\in\mathbb{R} , a\neq 0 \; \text{or} ...
4
votes
4answers
668 views

The relation between logic and algebra.

What's the relation between logic and algebra? Can one be thought of as a special case of the other?
2
votes
1answer
76 views

Which of the following representations of Karnaugh map is 'better'?

I usually come across two representations of Karnaugh maps in books and on the web as shown in the figure. The difference is whether the higher order variables are on the rows or on the columns. I ...
4
votes
2answers
207 views

What is the definition of a group entirely in first-order predicate logic?

I've seen the definition of a group in many different books given as follows: A group is a nonempty set $G$ and a binary operation $*$, denoted by $(G,*)$ that satisfies the following properties: ...
2
votes
2answers
175 views

Lefschetz Principle: explicit embeddings into $\mathbb C$.

I am very confused about the Lefschetz Principle. I read the Tarski Principle, but I am not acquainted with logic. Is there a statement more close to the language of field theory? Most of all, I ...
0
votes
0answers
141 views

Homomorphisms between structures [duplicate]

Given a structure $\mathcal{Y}$ and a family of structures $\left\{ \mathcal{A}_{i}:i\in I\right\} $, under what conditions is there a logical homomorphism between $\mathcal{Y}$ and $\mathcal{A}_{i}$ ...
2
votes
0answers
153 views

What is a good software package for ( assisted ) theorem proving and documenting?

Background: An issue in my math study is that I haven't found a good way of storing the theorems ( mostly abstract algebra ) I studied and want to (re-)use in proofs. At the moment I use a personal ...
2
votes
1answer
305 views

Injective Homomorphism and direct products

This question relates to products of structures all with the same symbol set $S$. After I give a little background the question follows. Direct Products This definition of the direct product is ...
1
vote
2answers
183 views

First order logic and algebraic structures. Reference request.

I have been encountering many results on logic-related properties of algebraic structures such as elementary equivalence, axiomatizability, definability, etc. The problem is that when I see the proof ...