# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthormatc
• CommentTimeJul 30th 2015
if "a partial function is the same as a functional relation seen from a different point of view." then why it is considered that partial functions are generalizations of functions, (at least that says Wikipedia in partial function entry) (if a function is "precisely a relation that is both functional and entire.")?
• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeJul 30th 2015

I don’t see a conflict between those two statements. Functions are special cases of partial functions: they are the ones that are entire as relations.

• CommentRowNumber3.
• CommentAuthorRodMcGuire
• CommentTimeJul 30th 2015
• (edited Jul 30th 2015)

“a partial function is the same as a functional relation seen from a different point of view.”

The understanding problem may involve the phrase “is the same”.

A partial function is a generalization of an “entire functional relation” in that it is not necessarily entire. However a partial function $f: A \to B$ is also equivalent to a total function to $B + \{\top\}$ which is a pointed set where the added point $\top$ is the “trash” point which stands for “undefined” or “unknown”. This added structure is a specialization.

Thus the relation between partial functions and total functions can be described as generalization, equivalence, and specialization. I expect the equivalence expresses a duality.

I think one could multi-point $B$ above with distinct points, $\top_1$, $\top_2 ...$ for each unknown value where the different $\top$s are isomorphic. Something like this is needed if one wants to capture the ordering of partial functions - if $f$ is a functional relation and $g = f + \{a \mapsto b\}$ augments $f$ with a new mapping that was undefined then $f \lt g$.

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeAug 27th 2015

Normally this extra point is called $\bot$ rather than $\top$ because it's at the bottom rather than the top of a partial order. This makes $f \leq g$ automatically work out in your example as you want: $f(a) = \bot \lt b = g(a)$. We should say more about this ordering at partial function.