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Results tagged with universal-algebra
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user 152679
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).
7
votes
1
answer
77
views
A syntactic characterisation of morphisms of algebraic theories whose induced algebraic func...
Let $f : S \to T$ be a morphism of algebraic theories. Such a morphism induces a monadic functor $f^* : \mathrm{Mod}(T) \to \mathrm{Mod}(S)$ (hence $f^*$ has a left adjoint). We may view $f$ syntactic …
- 15.9k
2
votes
Accepted
Nomenclature help: action vs. module and “pointed” monadic algebras are module objects?
Action versus module
In modern category theory parlance, action and module are typically interchangeable, and their use comes down to author preference: this is a historical artefact of the differing …
- 10.7k
2
votes
Accepted
Lawvere theory of Lawvere theories
Viewing cartesian operads as algebraic theories, your question may be rephrased as: "How can we present the (multisorted) algebraic theory whose models are (monosorted) algebraic theories?"
This is gi …
- 10.7k
7
votes
0
answers
188
views
Were algebraic theories and abstract clones defined independently?
Algebraic theories (by which I mean the formalism based on bijective-on-objects functors) and abstract clones both capture universal algebraic structure, and are well-known to be equivalent. Algebraic …
- 63.1k
8
votes
Comparing the existing formulations of universal algebra and their levels of generality
fosco gives a good overview of the general theory in their answer. I would like to point out two different generalisations of algebraic theory that are not, as far as I am aware, subsumed by any of th …
- 10.7k
2
votes
1
answer
112
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Characterisation of presentations for varietal large equational theories
Let $T : \mathbf{Set}^\mathrm{op} \to \mathscr T$ be a large equational theory (i.e. a bijective-on-objects product-preserving functor). Following Linton in Some Aspects of Equational Categories, we c …
- 64k
3
votes
Accepted
"Tietze-like transformations" for defining interesting bijections between algebraic structures
Tietze transformations for arbitrary algebraic theories (with respect to their presentations) have been considered in Malbos–Mimram's Homological Computations for Term Rewriting Systems, in the contex …
- 10.7k
8
votes
1
answer
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views
Why is the theory of small categories not algebraic?
In "Partial Horn logic and cartesian categories", E. Palmgren and S. J. Vickers state without proof that "The theory of categories is not algebraic." Is there a reference, or an elementary argument, f …
- 42.4k