The adjective "finite" applied to algebraic structures

In summary, the conversation discusses the usage of the adjective "finite" in algebraic structures. While it generally indicates a set with a finite number of elements, the term "finite algebra" specifically refers to finitely generated algebras. The question is raised whether there are other examples where "finite" means finite in some respect, but not necessarily as a set. The conversation also mentions the finite lattice representation problem, which asks whether finite lattices and finite algebras are the same. The examples provided show that "finite algebra" can refer to either a finite set or a finitely generated algebra, and it is important to clarify the meaning in each context.
  • #1
Stephen Tashi
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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?
 
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  • #2
Stephen Tashi said:
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.
 
  • #5
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.
 

FAQ: The adjective "finite" applied to algebraic structures

What does it mean for an algebraic structure to be "finite"?

When an algebraic structure is referred to as "finite", it means that it has a limited or finite number of elements. This is in contrast to an infinite algebraic structure, which has an infinite number of elements.

What are some examples of finite algebraic structures?

Examples of finite algebraic structures include finite groups, finite rings, and finite fields. These structures have a finite number of elements and follow specific rules and operations.

How does the concept of "finiteness" impact the properties of an algebraic structure?

The finiteness of an algebraic structure can greatly impact its properties. For example, finite algebraic structures have a finite number of substructures, which can make it easier to analyze and understand their properties. Additionally, certain properties, such as commutativity, may hold for all elements in a finite structure, whereas this may not be the case for infinite structures.

What are some applications of finite algebraic structures?

Finite algebraic structures have many practical applications, including in computer science, coding theory, and cryptography. They are also used in various branches of mathematics, such as abstract algebra and number theory.

Can an algebraic structure be both finite and infinite?

No, an algebraic structure cannot be both finite and infinite. The terms "finite" and "infinite" are mutually exclusive, meaning that a structure can only have one or the other characteristic. However, certain structures, such as the integers, can be considered both finite and infinite depending on the context in which they are being studied.

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