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Results tagged with universal-algebra
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user 44949
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).
2
votes
4
answers
549
views
relatively free groups in $Var(S_3)$
Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free?
This question is related to my previous question
Relatively free algebras in a variety …
- 63.9k
6
votes
2
answers
696
views
Non-trivial problems about the trivial group
Is there any non-trivial problem (maybe open problem) about the trivial group?
I asked already a question about the Laws characterizing the trivial group. There is a description of such laws. As ano …
CommunityBot
- 1
2
votes
1
answer
330
views
Algebras admitting quantifier elimination
I apologize if this question is meaningless or trivial:
What are examples of Algebras admitting quantifier elimination? Especially are there Groups admitting quantifier elimination?
I need to say so …
- 14.6k
1
vote
1
answer
143
views
Minimal generating sets of free algebras of varieties
Let $V$ be a variety and $F$ be a relatively free algebra in $V$. Suppose $X$ is a minimal generating set for $F$. Under what conditions we can deduce that $X$ is a free basis of $F$?
- 14.6k
1
vote
3
answers
380
views
Example of a non-finitely based variety with explicit set of defining identities
There are many examples of non-finitely based varieties. In a finite signature, is there an example of such variety with a known explicit set of identities?
- 563
3
votes
4
answers
906
views
Examples of algebras satisfying (a+b)(c+d)=ac+bd
Is there a known example of an algebra $(A, +, \cdot)$ with two binary commutative (see P.S below) and idempotent operations $+$ and $\cdot$ satisfying the identity $(a+b)(c+d)=ac+bd$?
Actually I …
13
votes
Accepted
Natural associative law for a ternary "group"?
What you need is the keyword "polyadic groups". A polyadic group is a non-empty set $G$ equipped with an associative $n$-ary operation $f:G^n\to G$ such that for all $a_1, \ldots, a_{n}$ and $b\in G$, …
- 2,233
0
votes
1
answer
193
views
Two questions about axiomatic rank of groups
Let $G$ be a group and $V=Var(G)$ be the variety generated by $G$. Suppose the axiomatic rank of $V$ is $n$. Let $Id(V)$ be the set of all identities of $V$.
1- Can we say that every element of $Id(V …
- 331
0
votes
1
answer
49
views
Product of two algebras with maximum condition
Suppose $A$ and $B$ are two algebras of the same signature, both having maximum condition on sub-algebras. Is it true that $A\times B$ has the same property?
- 38.6k
25
votes
1
answer
3k
views
A preprint of Sela concerning the work of Kharlampovich-Miyasnikov
Yesterday, Z. Sela published a preprint in arXiv which claims that the solution of Olga Kharlampovich and Alexi Miyasnikov for the Tarski problem on decidablity of the first order theories of free g …
- 2,233
0
votes
Are algebraic structures uniquely identifed by their free objects?
Two non-isomorphic algebras $A$ and $B$ may generate the same variety and so the free objects of the variety generated by an algebra $A$ do not characterize $A$ uniquely. For example, for any group $A …
- 2,233
4
votes
1
answer
205
views
Why the axiomatic rank of the variety of groups is equal to three?
I am thankful of Anton Klyachko who introduced axiomatic rank to me: the axiomatic rank of a variety is the minimum number of variables which we need to define that variety by identities.
It seems cl …
- 47.1k
3
votes
1
answer
126
views
Non finitely based varieties of groups defined by finitely many variables
A set $\Sigma$ of group identities is called bounded if there is $n\geq 1$ such that for any $(w\approx 1)\in \Sigma$, we have $w\in F(x_1, \ldots, x_n)$. A variety $\mathbf{V}$ is called bounded defi …
- 3,932
0
votes
3
answers
185
views
Negated varieties and their relatively free algebras
During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have …
CommunityBot
- 1
0
votes
Negated varieties and their relatively free algebras
This is a complete answer to the question 1, based on the idea of user45359:
We have $F_{Var^-(A)}(X)=T_{\mathcal{L}}(X)$.
Proof. Let $id^-(A)\vDash p\approx q$. By the unique readability of terms, …
- 2,233