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Results tagged with universal-algebra
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user 11640
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).
6
votes
Coequalizers in an Eilenberg-Moore category
This question is addressed by Linton [1969, Coequalizers in categories of algebras]. The first step is to notice that, in the presence of binary coproducts, coequalisers exist if and only if reflexive …
- 15.9k
16
votes
Accepted
Which categories are the categories of models of a Lawvere theory?
Adámek and Rosický [On sifted colimits and generalized varieties] have shown that a category $\mathcal{C}$ is equivalent to the category of models for a (finitary) Lawvere theory in $\mathbf{Set}$ if …
- 15.9k
7
votes
1
answer
333
views
What do algebraic theories with strictly terminal trivial models look like?
By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe uni …
- 15.9k
11
votes
1
answer
414
views
Examples of natural algebraic irreflexive relations
To motivate the question, consider the theory of rings.
Define $x \parallel y$ to mean $\exists w \exists z .((x - y) z = w (x - y) = 1)$, or in words, "$x - y$ is a unit".
Then $\parallel$ is a binar …
- 15.9k
7
votes
0
answers
130
views
Finitely presented algebras with isomorphic semilattices of congruences
Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward opera …
- 15.9k
23
votes
1
answer
962
views
Are there axioms satisfied in commutative rings and distributive lattices but not satisfied ...
Consider the language of rigs (also called semirings): it has constants $0$ and $1$ and binary operations $+$ and $\times$. The theory of commutative rigs is generated by the usual axioms: $+$ is asso …
- 15.9k
4
votes
Accepted
A syntactic characterisation of morphisms of algebraic theories whose induced algebraic func...
I thought about this a bit when trying to understand $\lambda$-rings – the same example that Wraith offers – and I think the answer is basically what is remarked in the cited lecture notes: it is nece …
- 15.9k