[Agda] algebra hierarchy in library

Sergei Meshveliani mechvel at botik.ru
Sat Mar 9 17:08:43 CET 2019


On Sat, 2019-03-09 at 17:08 +0300, Sergei Meshveliani wrote:
> On Sat, 2019-03-09 at 14:35 +0300, Sergei Meshveliani wrote:
> 
> > 
> > If this is the only reason, then it occurs that `Is' structure is needed
> > only in few cases.
> > For example, 
> > (1) There does not exist different semigroups that inherit ("are over")
> > the same Magma. The difference can be in a _proof_ for associativity,
> > but I doubt of whether this feature can be taken here in account. 
> > 
> > Let people correct me if I mistake in the following statements.
> > 
> > (2) There does not exist different monoids over the same Semigroup.
> > (3) There does not exist different commutative monoids over the same  
> >     Monoid.
> > (4) There does not exist different groups over the same Monoid.
> > (5) There does not exist different Abelian groups over the same Group.
> > (6) There does not exist different rings over the same Semiring.
> > (7) There does not exist different commutative rings over the same 
> >     Ring.
> [..]
> 
> 
> Sorry for a silly error. I discover a mistake in (2), (4) (6).
> 
> For example,  Nat1 = Nat\0  is a semigroup by _+_,  and zero can be
> joined in different ways, so that (Nat1 U 0) and (Nat1 U 0') occur
> different monoids. They are isomorphic, but they have different
> carriers. 
> A similar effect with carrier may be in (4) and (6).
> 
> So that there remain (1), (3) and (7).


No, again an error. 

A monoid in _this library_ is on the same carrier C as its inherited
semigroup. So that to implement a monoid on a given semigroup means to
choose any e in C which satisfy the law  \forall x (e*x == x*e == x)
and to prove this law.
And it is proved above that such  e  is unique.

So, I think that similarly, all the points (1) -- (7) are true
-- if I am not missing something.

Regards,

------
Sergei



> > 
> > > 
> > > On Fri, Mar 8, 2019 at 3:00 PM Sergei Meshveliani <mechvel at botik.ru>
> > > wrote:
> > > 
> > >         Dear standard library developers and supporters,
> > >         
> > >         can you please answer in (simple words) several questions
> > >         about the
> > >         representation of the algebraic hierarchy in standard library?
> > >         
> > >         
> > >         1. Why `Raw' structures?
> > >         
> > >         There are classical generic algebraic structures (call them
> > >         GAS):
> > >         Magma, Semigroup, Monoid, and so on.
> > >         
> > >         Those of them having some new field respectively to previous
> > >         structures
> > >         are accompanied with the corresponding `Raw' record. For
> > >         example, Magma
> > >         is preceded with RawMagma, Monoid with RawMonoid. Each `Raw'
> > >         structure
> > >         expresses only the signature of the corresponding GAS.
> > >         
> > >         What the `Raw' structures serve for?
> > >         
> > >         
> > >         2. Why putting `Is' structures into a different file?
> > >         
> > >         For example, the reader looks into Algebra.agda to find what
> > >         is
> > >         Semigroup:
> > >         
> > >         -------------------------------------------------------
> > >           record Semigroup c ℓ : Set (suc (c ⊔ ℓ)) where
> > >             ...
> > >             field Carrier     : Set c
> > >                   _≈_         : Rel Carrier ℓ
> > >                   _∙_         : Op₂ Carrier
> > >                   isSemigroup : IsSemigroup _≈_ _∙_
> > >         
> > >             open IsSemigroup isSemigroup public
> > >             ...
> > >             magma = record { isMagma = isMagma }
> > >             ...
> > >         
> > >         Now, one needs to find a declaration for IsSemigroup.
> > >         And it resides in a different file of
> > >                                            Algebra/Structures.agda :
> > >         
> > >           record IsSemigroup (∙ : Op₂ A) : Set (a ⊔ ℓ) where
> > >             field
> > >               isMagma : IsMagma ∙
> > >               assoc   : Associative ∙
> > >         
> > >             open IsMagma isMagma public
> > >         ---------------------------------------------------------
> > >         
> > >         And all this implements the meaning of a small sentence:
> > >         ``Semigroup is Magma in which multiplication _∙_ is
> > >         associative''.
> > >         
> > >         Why not put ``record IsSemigroup'' before ``record Semigroup''
> > >         in the
> > >         same file  Algebra.agda ?
> > >         
> > >         Similarly other `Is' GAS decls can join. So that Algebra.agda
> > >         and
> > >         Algebra/Structures.agda will merge into  Algebra.agda   in
> > >         which each
> > >         GAS will be defined in one place.
> > >         For example, to see what is a group will need to look into one
> > >         file, not
> > >         in two files.
> > >         ?
> > >         
> > >         
> > >         3. Why Magma declares the fields  Carrier and _≈_  by new?
> > >         
> > >         Similarly, why other GAS re-declare many fields?
> > >         
> > >         In theory, we have
> > >         ``Magma is a setoid with an operation _∙_ congruent with
> > >         respect to the
> > >         equality _≈_''.
> > >         
> > >         So, Magma inherits Setoid. And it is natural for its
> > >         representation in
> > >         Agda to have  setoid  somewhere inside it. So, it opens this
> > >         setoid and
> > >         uses its fields in further definitions. For example, like
> > >         this:
> > >         
> > >         -- Version II
> > >         ---------------------------------------------------------
> > >         
> > >         record IsMagma {α α=} (S : Setoid α α=) (_∙_ : Op₂
> > >         (Setoid.Carrier S)) :
> > >                                                                    Set
> > >         (α ⊔ α=)
> > >           where
> > >           open Setoid S using (_≈_; Carrier)
> > >           field
> > >             ∙cong : _∙_ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_
> > >         
> > >         record Magma α α= :  Set (suc (α ⊔ α=))
> > >           where
> > >           field  setoid : Setoid α α=
> > >         
> > >           open Setoid setoid public
> > >           infixl 7 _∙_
> > >         
> > >           field  _∙_     : Op₂ Carrier
> > >                  isMagma : IsMagma setoid _∙_
> > >         ------------------------------------------------------------------------
> > >         
> > >         And let us call  Version I  the approach of Standard library
> > >         lib-0.17.
> > >         
> > >         Both versions use an `Is' structure, but II does not
> > >         re-declare fields.
> > >         Is not II more natural?
> > >         
> > >         
> > >         Another question may be:
> > >         ``why splitting each GAS into proper structure and `Is'
> > >         structure?''.
> > >         
> > >         My guess is that this approach allows us to express two GAS-s
> > >         that are
> > >         over the same inherited GAS. For example, a programmer can
> > >         express a
> > >         product of two Magmae over the same Setoid:
> > >         ------------------------------------------------------------
> > >         module _ {α α=} (S : Setoid α α=)
> > >           where
> > >           open Setoid S using (Carrier; _≈_)
> > >           SS = ×-setoid S S
> > >           open Setoid SS using () renaming (Carrier to CC; _≈_ to
> > >         _=p_)
> > >         
> > >           magmaProduct' :
> > >             (_*₁_ _*₂_ : Op₂ Carrier) → IsMagma _≈_ _*₁_ →
> > >                                         IsMagma _≈_ _*₂_ → Magma α α=
> > >           magmaProduct' _*₁_ _*₂_ insM₁ isM₂ =
> > >                                <define coordinate-wise multiplcation
> > >         on CC;
> > >                                 prove ...;  return the Magma record
> > >                                >
> > >         ------------------------------------------------------------
> > >         
> > >         (is there any other purpose to introduce `Is' -structures?).
> > >         
> > >         This is equally easy to set both in Version I and Version II.
> > >         
> > >         But note that both approaches still deviate, a bit, from the
> > >         theory.
> > >         Because in theory, it is 
> > >         \ (mg1 : Magma_ _) (mg2 : Magma _ _) (HaveCommonSetoid mg1
> > >         mg2) →
> > >         product-magma,
> > >         while magmaProduct' takes certain parts of the two magmae.
> > >         
> > >         So, there remain somewhat three and a half questions.
> > >         
> > >         Thank you in advance for your possible explanation.
> > >         
> > >         ------
> > >         Sergei
> > >         
> > >         
> > >         
> > >         
> > >         _______________________________________________
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> > >         Agda at lists.chalmers.se
> > >         https://lists.chalmers.se/mailman/listinfo/agda
> > 
> > 
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