Re: Notions of Type
From: paul c <toledobythesea_at_oohay.ac>
Date: Thu, 17 Aug 2006 01:12:16 GMT
Message-ID: <QnPEg.410590$iF6.320822_at_pd7tw2no>
...
> My only objection to "type = algebraic structure" is the requirement for
> closure within a single type. For any type, we can define the algebra as
> the type and the subset of the type's operations that exhibit closure.
> In my view, the type includes the entire set of operations defined on
> the values of the type including those not exhibiting closure.
>
> For instance, an operation might have a single character string operand
> with non-negative integers as result as is the case with the length
> operation. That operation is certainly part of the character string type
> and arguably part of the non-negative integer type. It is not part of
> any algebraic structure. Or do I misunderstand something?
>
> Square root is part of the algebra for non-negative reals and for
> perfect squares but not for integers. It is, however, a valid operation
> on integers resulting in values of a different type.
>
> Division is not part of the algebra for any type that includes zero as
> one of its values. Or is it?
>
> Other than possibly the _Principle of Cautious Design_, I can think of
> no immediate objection to a type where the set of operations in the
> algebra is empty. On the other hand, it might be difficult to devise
> such a type that has any use.
Date: Thu, 17 Aug 2006 01:12:16 GMT
Message-ID: <QnPEg.410590$iF6.320822_at_pd7tw2no>
Bob Badour wrote:
> Keith H Duggar wrote:
>
>> Bob Badour wrote: >> >>> JOG wrote: >>>
...
> My only objection to "type = algebraic structure" is the requirement for
> closure within a single type. For any type, we can define the algebra as
> the type and the subset of the type's operations that exhibit closure.
> In my view, the type includes the entire set of operations defined on
> the values of the type including those not exhibiting closure.
>
> For instance, an operation might have a single character string operand
> with non-negative integers as result as is the case with the length
> operation. That operation is certainly part of the character string type
> and arguably part of the non-negative integer type. It is not part of
> any algebraic structure. Or do I misunderstand something?
>
> Square root is part of the algebra for non-negative reals and for
> perfect squares but not for integers. It is, however, a valid operation
> on integers resulting in values of a different type.
>
> Division is not part of the algebra for any type that includes zero as
> one of its values. Or is it?
>
> Other than possibly the _Principle of Cautious Design_, I can think of
> no immediate objection to a type where the set of operations in the
> algebra is empty. On the other hand, it might be difficult to devise
> such a type that has any use.
p Received on Thu Aug 17 2006 - 03:12:16 CEST