Characterize Powersets among CPOs
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Powersets can be seen as complete atomic boolean algebras.
Is it possible to characterize complete atomic boolean algebras (CABA) among complete partial orders (CPO)?
For example, it is possible to characterize complete Heyting Algebras as complete posets where $$a wedge bigvee a_i = bigvee a wedge a_i.$$
In this sense a complete boolen algebra is just a complete partial order with an exactness property, the infinitary distributivity law.
My guess, in fact my hope, is that CABAs correspond to complete atomic regular posets. By regular I mean that the poset is a regular category.
category-theory order-theory boolean-algebra lattice-orders
asked Aug 7 at 15:52
Ivan Di Liberti
2,29311122
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up vote
2
down vote
favorite
Powersets can be seen as complete atomic boolean algebras.
Is it possible to characterize complete atomic boolean algebras (CABA) among complete partial orders (CPO)?
For example, it is possible to characterize complete Heyting Algebras as complete posets where $$a wedge bigvee a_i = bigvee a wedge a_i.$$
In this sense a complete boolen algebra is just a complete partial order with an exactness property, the infinitary distributivity law.
My guess, in fact my hope, is that CABAs correspond to complete atomic regular posets. By regular I mean that the poset is a regular category.
category-theory order-theory boolean-algebra lattice-orders
asked Aug 7 at 15:52
Ivan Di Liberti
2,29311122
1
I don't understand what you mean by "characterize" ? Their definition is entirely in terms of order-theoretic notions so I think you'll have to be more precise
– Max
Aug 7 at 15:56
As I say in the question, complete HAs can be characterized as complete and cartesian closed poset. I am looking for a similar characterization that only involves property-like structure.
– Ivan Di Liberti
Aug 7 at 16:14
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Powersets can be seen as complete atomic boolean algebras.
Is it possible to characterize complete atomic boolean algebras (CABA) among complete partial orders (CPO)?
For example, it is possible to characterize complete Heyting Algebras as complete posets where $$a wedge bigvee a_i = bigvee a wedge a_i.$$
In this sense a complete boolen algebra is just a complete partial order with an exactness property, the infinitary distributivity law.
My guess, in fact my hope, is that CABAs correspond to complete atomic regular posets. By regular I mean that the poset is a regular category.
category-theory order-theory boolean-algebra lattice-orders
asked Aug 7 at 15:52
Ivan Di Liberti
2,29311122
Powersets can be seen as complete atomic boolean algebras.
Is it possible to characterize complete atomic boolean algebras (CABA) among complete partial orders (CPO)?
For example, it is possible to characterize complete Heyting Algebras as complete posets where $$a wedge bigvee a_i = bigvee a wedge a_i.$$
In this sense a complete boolen algebra is just a complete partial order with an exactness property, the infinitary distributivity law.
My guess, in fact my hope, is that CABAs correspond to complete atomic regular posets. By regular I mean that the poset is a regular category.
category-theory order-theory boolean-algebra lattice-orders
asked Aug 7 at 15:52
Ivan Di Liberti
2,29311122
edited Aug 7 at 19:39
asked Aug 7 at 15:52
Ivan Di Liberti
2,29311122
asked Aug 7 at 15:52
Ivan Di Liberti
2,29311122
asked Aug 7 at 15:52
Ivan Di Liberti
2,29311122
2,29311122
1
I don't understand what you mean by "characterize" ? Their definition is entirely in terms of order-theoretic notions so I think you'll have to be more precise
– Max
Aug 7 at 15:56
As I say in the question, complete HAs can be characterized as complete and cartesian closed poset. I am looking for a similar characterization that only involves property-like structure.
– Ivan Di Liberti
Aug 7 at 16:14
add a comment |Â
1
I don't understand what you mean by "characterize" ? Their definition is entirely in terms of order-theoretic notions so I think you'll have to be more precise
– Max
Aug 7 at 15:56
As I say in the question, complete HAs can be characterized as complete and cartesian closed poset. I am looking for a similar characterization that only involves property-like structure.
– Ivan Di Liberti
Aug 7 at 16:14
1
1
I don't understand what you mean by "characterize" ? Their definition is entirely in terms of order-theoretic notions so I think you'll have to be more precise
– Max
Aug 7 at 15:56
I don't understand what you mean by "characterize" ? Their definition is entirely in terms of order-theoretic notions so I think you'll have to be more precise
– Max
Aug 7 at 15:56
As I say in the question, complete HAs can be characterized as complete and cartesian closed poset. I am looking for a similar characterization that only involves property-like structure.
– Ivan Di Liberti
Aug 7 at 16:14
As I say in the question, complete HAs can be characterized as complete and cartesian closed poset. I am looking for a similar characterization that only involves property-like structure.
– Ivan Di Liberti
Aug 7 at 16:14
add a comment |Â
1 Answer
1
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oldest
votes
up vote
2
down vote
accepted
Assuming by complete you mean in the categorical sense, i.e. you want limit-closure (i.e. meet-closure in the poset case), I suppose that your complete boolean algebras should be the same as complete star-autonomous categories where the monoidal product is the cartesian one.
In these categories you have an involution operator (the $*$ in star-autonomous) which plays the role of the negation and indeed you have, in the posetal case, de morgan laws and the middle excluded.
Edit: I see thatthe OP was looking for a property-like definition, hence the following addendum.
As shown in the link above $*$-autonomous categories admit a definition in a almost-property like way (as long as we are considering just cartesian closed category) using the dualizing object $(bot)$ instead of the involution functor $(*)$.
This is an almost property like, as opposed to property like, because we require the object $bot$ to satisfy the condition
the morphism $A to ((A multimap bot) multimap bot)$ to be an isomorphism
the problem is that in a general category this object would be an additional structure on the category.
But if our cartesian closed category, $*$-auotnomous, is a poset $bot$ is necessarily the bottom element, i.e. the minimum.
To see that one can observe that if $ ((A multimap bot) multimap bot) leq A$, which is the poset-version of the property above, since obviously $bot leq ((A multimap bot) multimap bot)$ it follows that $bot leq A$.
From this it follows that in a star-autonomous (cartesian) poset the global dualizing object must be the minimum of the poset.
So a star-autonomous-poset is nothing but a cartesian-closed-posetal category with a minimum $bot$ that satisfies the property
$$((A multimap bot) multimap bot leq A$$
This should provide a property-like definition as you wished....or at least I hope so :-D
answered Aug 7 at 16:33
Giorgio Mossa
13.3k11748
1
I am sorry to say that this is not the kind of answer I am looking for. In fact the $*$ operator is an additional structure, and not a property. In particular the operator $neg$ is perfectly definable in a complete lattice as $neg a := bigvee_d : d wedge a = bot d$. Thus one could state the property of "having the $neg$ operator" as an exactness property. What I really want to know is if this exactness property coincides with a very classical one, such as regularity, or being Heyting or whatever.
– Ivan Di Liberti
Aug 7 at 17:54
@IvanDiLiberti I've made a little addendum that should address your concerns. Let me know if it's ok.
– Giorgio Mossa
Aug 7 at 19:24
Thanks, this is perfect.
– Ivan Di Liberti
Aug 8 at 5:33
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Assuming by complete you mean in the categorical sense, i.e. you want limit-closure (i.e. meet-closure in the poset case), I suppose that your complete boolean algebras should be the same as complete star-autonomous categories where the monoidal product is the cartesian one.
In these categories you have an involution operator (the $*$ in star-autonomous) which plays the role of the negation and indeed you have, in the posetal case, de morgan laws and the middle excluded.
Edit: I see thatthe OP was looking for a property-like definition, hence the following addendum.
As shown in the link above $*$-autonomous categories admit a definition in a almost-property like way (as long as we are considering just cartesian closed category) using the dualizing object $(bot)$ instead of the involution functor $(*)$.
This is an almost property like, as opposed to property like, because we require the object $bot$ to satisfy the condition
the morphism $A to ((A multimap bot) multimap bot)$ to be an isomorphism
the problem is that in a general category this object would be an additional structure on the category.
But if our cartesian closed category, $*$-auotnomous, is a poset $bot$ is necessarily the bottom element, i.e. the minimum.
To see that one can observe that if $ ((A multimap bot) multimap bot) leq A$, which is the poset-version of the property above, since obviously $bot leq ((A multimap bot) multimap bot)$ it follows that $bot leq A$.
From this it follows that in a star-autonomous (cartesian) poset the global dualizing object must be the minimum of the poset.
So a star-autonomous-poset is nothing but a cartesian-closed-posetal category with a minimum $bot$ that satisfies the property
$$((A multimap bot) multimap bot leq A$$
This should provide a property-like definition as you wished....or at least I hope so :-D
answered Aug 7 at 16:33
Giorgio Mossa
13.3k11748
1
I am sorry to say that this is not the kind of answer I am looking for. In fact the $*$ operator is an additional structure, and not a property. In particular the operator $neg$ is perfectly definable in a complete lattice as $neg a := bigvee_d : d wedge a = bot d$. Thus one could state the property of "having the $neg$ operator" as an exactness property. What I really want to know is if this exactness property coincides with a very classical one, such as regularity, or being Heyting or whatever.
– Ivan Di Liberti
Aug 7 at 17:54
@IvanDiLiberti I've made a little addendum that should address your concerns. Let me know if it's ok.
– Giorgio Mossa
Aug 7 at 19:24
Thanks, this is perfect.
– Ivan Di Liberti
Aug 8 at 5:33
add a comment |Â
up vote
2
down vote
accepted
Assuming by complete you mean in the categorical sense, i.e. you want limit-closure (i.e. meet-closure in the poset case), I suppose that your complete boolean algebras should be the same as complete star-autonomous categories where the monoidal product is the cartesian one.
In these categories you have an involution operator (the $*$ in star-autonomous) which plays the role of the negation and indeed you have, in the posetal case, de morgan laws and the middle excluded.
Edit: I see thatthe OP was looking for a property-like definition, hence the following addendum.
As shown in the link above $*$-autonomous categories admit a definition in a almost-property like way (as long as we are considering just cartesian closed category) using the dualizing object $(bot)$ instead of the involution functor $(*)$.
This is an almost property like, as opposed to property like, because we require the object $bot$ to satisfy the condition
the morphism $A to ((A multimap bot) multimap bot)$ to be an isomorphism
the problem is that in a general category this object would be an additional structure on the category.
But if our cartesian closed category, $*$-auotnomous, is a poset $bot$ is necessarily the bottom element, i.e. the minimum.
To see that one can observe that if $ ((A multimap bot) multimap bot) leq A$, which is the poset-version of the property above, since obviously $bot leq ((A multimap bot) multimap bot)$ it follows that $bot leq A$.
From this it follows that in a star-autonomous (cartesian) poset the global dualizing object must be the minimum of the poset.
So a star-autonomous-poset is nothing but a cartesian-closed-posetal category with a minimum $bot$ that satisfies the property
$$((A multimap bot) multimap bot leq A$$
This should provide a property-like definition as you wished....or at least I hope so :-D
answered Aug 7 at 16:33
Giorgio Mossa
13.3k11748
1
I am sorry to say that this is not the kind of answer I am looking for. In fact the $*$ operator is an additional structure, and not a property. In particular the operator $neg$ is perfectly definable in a complete lattice as $neg a := bigvee_d : d wedge a = bot d$. Thus one could state the property of "having the $neg$ operator" as an exactness property. What I really want to know is if this exactness property coincides with a very classical one, such as regularity, or being Heyting or whatever.
– Ivan Di Liberti
Aug 7 at 17:54
@IvanDiLiberti I've made a little addendum that should address your concerns. Let me know if it's ok.
– Giorgio Mossa
Aug 7 at 19:24
Thanks, this is perfect.
– Ivan Di Liberti
Aug 8 at 5:33
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Assuming by complete you mean in the categorical sense, i.e. you want limit-closure (i.e. meet-closure in the poset case), I suppose that your complete boolean algebras should be the same as complete star-autonomous categories where the monoidal product is the cartesian one.
In these categories you have an involution operator (the $*$ in star-autonomous) which plays the role of the negation and indeed you have, in the posetal case, de morgan laws and the middle excluded.
Edit: I see thatthe OP was looking for a property-like definition, hence the following addendum.
As shown in the link above $*$-autonomous categories admit a definition in a almost-property like way (as long as we are considering just cartesian closed category) using the dualizing object $(bot)$ instead of the involution functor $(*)$.
This is an almost property like, as opposed to property like, because we require the object $bot$ to satisfy the condition
the morphism $A to ((A multimap bot) multimap bot)$ to be an isomorphism
the problem is that in a general category this object would be an additional structure on the category.
But if our cartesian closed category, $*$-auotnomous, is a poset $bot$ is necessarily the bottom element, i.e. the minimum.
To see that one can observe that if $ ((A multimap bot) multimap bot) leq A$, which is the poset-version of the property above, since obviously $bot leq ((A multimap bot) multimap bot)$ it follows that $bot leq A$.
From this it follows that in a star-autonomous (cartesian) poset the global dualizing object must be the minimum of the poset.
So a star-autonomous-poset is nothing but a cartesian-closed-posetal category with a minimum $bot$ that satisfies the property
$$((A multimap bot) multimap bot leq A$$
This should provide a property-like definition as you wished....or at least I hope so :-D
answered Aug 7 at 16:33
Giorgio Mossa
13.3k11748
Assuming by complete you mean in the categorical sense, i.e. you want limit-closure (i.e. meet-closure in the poset case), I suppose that your complete boolean algebras should be the same as complete star-autonomous categories where the monoidal product is the cartesian one.
In these categories you have an involution operator (the $*$ in star-autonomous) which plays the role of the negation and indeed you have, in the posetal case, de morgan laws and the middle excluded.
Edit: I see thatthe OP was looking for a property-like definition, hence the following addendum.
As shown in the link above $*$-autonomous categories admit a definition in a almost-property like way (as long as we are considering just cartesian closed category) using the dualizing object $(bot)$ instead of the involution functor $(*)$.
This is an almost property like, as opposed to property like, because we require the object $bot$ to satisfy the condition
the morphism $A to ((A multimap bot) multimap bot)$ to be an isomorphism
the problem is that in a general category this object would be an additional structure on the category.
But if our cartesian closed category, $*$-auotnomous, is a poset $bot$ is necessarily the bottom element, i.e. the minimum.
To see that one can observe that if $ ((A multimap bot) multimap bot) leq A$, which is the poset-version of the property above, since obviously $bot leq ((A multimap bot) multimap bot)$ it follows that $bot leq A$.
From this it follows that in a star-autonomous (cartesian) poset the global dualizing object must be the minimum of the poset.
So a star-autonomous-poset is nothing but a cartesian-closed-posetal category with a minimum $bot$ that satisfies the property
$$((A multimap bot) multimap bot leq A$$
This should provide a property-like definition as you wished....or at least I hope so :-D
answered Aug 7 at 16:33
Giorgio Mossa
13.3k11748
edited Aug 7 at 19:23
answered Aug 7 at 16:33
Giorgio Mossa
13.3k11748
answered Aug 7 at 16:33
Giorgio Mossa
13.3k11748
answered Aug 7 at 16:33
Giorgio Mossa
13.3k11748
13.3k11748
1
I am sorry to say that this is not the kind of answer I am looking for. In fact the $*$ operator is an additional structure, and not a property. In particular the operator $neg$ is perfectly definable in a complete lattice as $neg a := bigvee_d : d wedge a = bot d$. Thus one could state the property of "having the $neg$ operator" as an exactness property. What I really want to know is if this exactness property coincides with a very classical one, such as regularity, or being Heyting or whatever.
– Ivan Di Liberti
Aug 7 at 17:54
@IvanDiLiberti I've made a little addendum that should address your concerns. Let me know if it's ok.
– Giorgio Mossa
Aug 7 at 19:24
Thanks, this is perfect.
– Ivan Di Liberti
Aug 8 at 5:33
add a comment |Â
1
I am sorry to say that this is not the kind of answer I am looking for. In fact the $*$ operator is an additional structure, and not a property. In particular the operator $neg$ is perfectly definable in a complete lattice as $neg a := bigvee_d : d wedge a = bot d$. Thus one could state the property of "having the $neg$ operator" as an exactness property. What I really want to know is if this exactness property coincides with a very classical one, such as regularity, or being Heyting or whatever.
– Ivan Di Liberti
Aug 7 at 17:54
@IvanDiLiberti I've made a little addendum that should address your concerns. Let me know if it's ok.
– Giorgio Mossa
Aug 7 at 19:24
Thanks, this is perfect.
– Ivan Di Liberti
Aug 8 at 5:33
1
1
I am sorry to say that this is not the kind of answer I am looking for. In fact the $*$ operator is an additional structure, and not a property. In particular the operator $neg$ is perfectly definable in a complete lattice as $neg a := bigvee_d : d wedge a = bot d$. Thus one could state the property of "having the $neg$ operator" as an exactness property. What I really want to know is if this exactness property coincides with a very classical one, such as regularity, or being Heyting or whatever.
– Ivan Di Liberti
Aug 7 at 17:54
I am sorry to say that this is not the kind of answer I am looking for. In fact the $*$ operator is an additional structure, and not a property. In particular the operator $neg$ is perfectly definable in a complete lattice as $neg a := bigvee_d : d wedge a = bot d$. Thus one could state the property of "having the $neg$ operator" as an exactness property. What I really want to know is if this exactness property coincides with a very classical one, such as regularity, or being Heyting or whatever.
– Ivan Di Liberti
Aug 7 at 17:54
@IvanDiLiberti I've made a little addendum that should address your concerns. Let me know if it's ok.
– Giorgio Mossa
Aug 7 at 19:24
@IvanDiLiberti I've made a little addendum that should address your concerns. Let me know if it's ok.
– Giorgio Mossa
Aug 7 at 19:24
Thanks, this is perfect.
– Ivan Di Liberti
Aug 8 at 5:33
Thanks, this is perfect.
– Ivan Di Liberti
Aug 8 at 5:33
add a comment |Â
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1
I don't understand what you mean by "characterize" ? Their definition is entirely in terms of order-theoretic notions so I think you'll have to be more precise
– Max
Aug 7 at 15:56
As I say in the question, complete HAs can be characterized as complete and cartesian closed poset. I am looking for a similar characterization that only involves property-like structure.
– Ivan Di Liberti
Aug 7 at 16:14