> What is the prevailing sentiment today among readers of this list
> concerning this question? Do you feel that every variety should be
> allowed its initial algebra, or is your position that the varieties
> without constants in the signature should not be allowed an initial
> algebra? And if the latter, why? Because there is something
> technically or morally wrong with empty algebras, or because it is far
> too late now to change the well-established definition of variety, or
> some other reason?

The situation above is similar to the question whether the zero and the
units are irreducibles (or primes) in Z by definition or not.

The mankind has for candidates for the "right" definition, according to
the cases whether the zero or the units are accepted as irreducibles.

So what makes the decision?

Widening the question: introducing a new notion (i.e. an abbreviation),
what makes the possible conservative extension of the theory in question,
a popular and widely used?

The underlying language is also extended by the conservative extension,
and usually the extended language yields a better compression of the
knowledge encoded using the extended language in question.

Not necessarily the compression ratio makes the final decision, since the
handiness of the abbreviation does matter as well.

The better compressions (i.e. better notions) provide better winning
position in the evolution of the knowledge, since easier to generate new
results, to understand the connections, to get a better overview.

In artificial intelligence experiments one can observe the phenomenon,
that whenever the overall size of the current knowledge is just collapsed,
a new essential idea, lemma, notion can be found in the knowledge
collected or generated by the system.

It is a matter of fact, that a book on the introduction to the number
theory is shorter and easyer to understand, if the zero and the units are
not considered irreducible. And that's all!

There are four questions concerning the "opinion poll":

What does all of the above mean for the question of

1: initial objects in category theory?

2: initial objects in universal algebra?

3: empty algebras in category theory?

4: empty algebras in universal algebra?

First, the two obvious cases:

1: initial object can occur in category theory, say in category created on
the base of a universal algebraic variety. Initial object a successful
notion, if you have a category theoretical question.

4: empty algebras never occur in universal algebra, simply because the
compression appeared to be better, if the empty algebras are forbidden.
Empty algebra is not too useful notion, if you have a universal algebraic
question.

Note: the famous and very important AXIOM OF CHOICE is nothing else, but a
specific VERY COMMON algebra construction DOES NEVER OUTPUT AN EMPTY
ALGEBRA!

Second, the two less obvious cases:

2+3: these problems are similar, namely: what to do with the foreign
notions imported from other areas of mathematics?

Chaos can be established, if the MEANING of the notions are altered along
their import from other areas.

What is the real reason for the imports in question?

Probably a result of the other area is needed.

Say, in the case "2:", the sound procedure is, considering a universal
algebraic variety, first to create the corresponding category, second to
extend that category by an initial object, then to apply the category
theoretical theorem for that extended category, and finally to derive the
meaning for the original variety, while the case and influence of the
initial object must be translated back correctly.

Peter.