I The adjective "finite" applied to algebraic structures

[Stephen Tashi]

TL;DR Summary

Are there many examples in algebra where the adjective "finite" (by itself) means "finitely generated" or "finite dimensional" or finite in some other sense than being a finite set?

The adjective "finite" applied to many algebraic structures (e.g. groups, fields) indicates a set with a finite number of elements. However, (as I understand it) "finite algebra" refers to a finitely generated algebra. Are there other examples where "finite" means finite in some respect but not necessarily finite as a set?

 

[fresh_42]

Summary:: Are there many examples in algebra where the adjective "finite" (by itself) means "finitely generated" or "finite dimensional" or finite in some other sense than being a finite set?

The adjective "finite" applied to many algebraic structures (e.g. groups, fields) indicates a set with a finite number of elements. However, (as I understand it) "finite algebra" refers to a finitely generated algebra. Are there other examples where "finite" means finite in some respect but not necessarily finite as a set?

I wouldn't call a finitely generated algebra just finite. This is misleading, as it could mean that the set of elements of the algebra is finite. There is a reason why it is called finitely generated, in which case the set of generators is finite. So in any case there is some finite set if we use this adjective, be it generators, basis elements, or the entire set.

Your question doesn't make much sense to me.

 

[Stephen Tashi]

Your question doesn't make much sense to me.

Look at https://en.wikipedia.org/wiki/Finite_lattice_representation_problem

The finite lattice representation problem asks whether the same is true for finite lattices and finite algebras.

Is a "finite algebra" a finite set?

 

[Stephen Tashi]

Example of "finite algebra" as meaning finitely generated:
https://en.wikipedia.org/wiki/Finite_algebra
- or is the first sentence of that article a theorem instead of a definition?

Example of "finite algebra" as a finite set
http://www.math.hawaii.edu/~williamdemeo/latticetheory/PalfyPudlakTalk.pdf

 

[fresh_42]

I assume it should better be "finite dimensional" in both cases, lattice and algebra. But since I have never heard, nor do I have an imagination of a finite lattice, it could as well be finite sets. In that case it will inevitably imply a finite field.

The quotation 'Intervals in subgroup lattices of finite groups.' indicates finite sets. We have two different words, Gitter = lattice and Verband = lattice order. So it seems we are talking about lattice orders here. The question is thus whether such a (finite) lattice (order) can be written as a quotient algebra. Now as a quotient it could mean finite algebras or finite dimensional algebras, because the quotient is a set of equivalence classes and the word finite alone does not indicate what is canceled out. I assume, however, that actually finite algebras (finite set of elements) are meant, since otherwise we would say finite dimensional or finite generated. I would search for and look into the original paper in this case to be clear.