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A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms).

Boolean Algebras that are complete as well as atomic (also called CABAs) are of course precisely those that are isomorphic to some power set (equipped with the obvious choices for the operations), or equivalently stated, those that form a category dually equivalent to $Set$.

The category of all Boolean algbras, however, is well-known to be equivalent to the category of Stone spaces (compact totally disconnected Hausdorff spaces) with continuous morphisms. Thus, for a Boolean algebra (of infinite cardinality), it is a very special case to be complete and atomic. My question is:

What are nice examples for Boolean Algebras that are not complete or not atomic?

Please understand that I do not look for any kind of example (so the emphasis lies on the word "nice"). I am, for instance, aware that looking at free BAs would lead to such an example, and I also know the classic example of the BA that is formed by all finite and confinite sets of integers. Also, as mentioned above, I know how the Stone Duality workstransforms Stone spaces into Boolean algebras, so please don't simply say "the clopen subsets of a that-and-that Stone-Space form a Boolean algebra".

I admit that nice is a somewhat vague notion. What I mean are Boolean Algebras that arise naturally (except those I have already mentioned) and are of special interest for some reason (yes, I know that this formulation is not vague at all).

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A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms).

Boolean Algebras that are complete as well as atomic (also called CABAs) are of course precisely those that are isomorphic to some power set (equipped with the obvious choices for the operations), or equivalently stated, those that form a category dually equivalent to $Set$.

The category of all Boolean algbras, however, is well-known to be equivalent to the category of Stone spaces (compact totally disconnected Hausdorff spaces) with continuous morphisms. Described in this wayThus, the for a Boolean algebras that are not atomic or not complete are precisely those that are carried algebra (of infinite cardinality), it is a very special case to non-discrete Stone spaces via the Stone Dualitybe complete and atomic. My question is:

What are nice examples for such Boolean Algebras that are not complete or not atomic?

Please understand that I do not look for any kind of example (so the emphasis lies on the word "nice"). I am, for instance, aware that looking at free BAs would lead to such an example, and I also know the classic example of the BA that is formed by all finite and confinite sets of integers. Also, as mentioned above, I know how the Stone Duality works, so please don't simply say "the clopen subsets of a that-and-that Stone-Space form a Boolean algebra".

I admit that nice is a somewhat vague notion. What I mean are Boolean Algebras that arise naturally (except those I have already mentioned) and are of special interest for some reason (yes, I know that this formulation is not vague at all).

A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms).

Boolean Algebras that are complete as well as atomic (also called CABAs) are of course precisely those that are isomorphic to some power set (equipped with the obvious choices for the operations), or equivalently stated, those that form a category dually equivalent to $Set$.

The category of all Boolean algbras, however, is well-known to be equivalent to the category of Stone spaces (compact totally disconnected Hausdorff spaces) with continuous morphisms. Described in this way, the Boolean algebras that are not atomic or not complete are precisely those that are carried to non-discrete Stone spaces via the Stone Duality. My question is:

What are nice examples for such Boolean Algebras?

Please understand that I do not look for any kind of example (so the emphasis lies on the word "nice"). I am, for instance, aware that looking at free BAs would lead to such an example, and I also know the classic example of the BA that is formed by all finite and confinite sets of integers. Also, as mentioned above, I know how the Stone Duality works, so please don't simply say "the clopen subsets of a that-and-that Stone-Space form a Boolean algebra".

I admit that nice is a somewhat vague notion. What I mean are Boolean Algebras that arise naturally (except those I have already mentioned) and are of special interest for some reason (yes, I know that this formulation is not vague at all).

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