Hi, Vaughan,

The universal algebra I learned came out of Slominski's
little tract on infinitary operations.

For him what was of interest were "equationally definable
classes of algebras." And for him "algebras" had no need
to be non-empty. Thus for him there were always free algebras
on any sets of generators, even empty ones, even if they
turned out (as when no zero-ary operations are at hand)
to be empty. Same, therefore, for me.

Cheers,

-- Fred

------ Original Message ------
Received: Sat, 22 Sep 2007 08:55:11 PM EDT
From: Vaughan Pratt <pratt@...>
To: univalg@yahoogroups.com
Subject: [univalg] Opinion poll: does every variety have an initial object

> It is a theorem of category theory, but not of universal algebra, that
> every variety has an initial algebra.
>
> In category theory the initial algebra can be found equivalently either
> in the Kleisli category of a monad (consisting of the free algebras of
> the variety) as the free algebra on no generators, or in the
> Eilenberg-Moore category of that monad (consisting of the whole variety)
> as the algebra whose underlying set consists of the constants.
>
> In universal algebra on the other hand, none of the varieties of
> semigroups, semilattices, lattices, modular lattices, and distributive
> lattices as defined in McKenzie, McNulty, and Taylor have an initial
> object, by fiat: empty algebras are forbidden.
>
> This was a source of some controversy at the Universal Algebra and
> Category Theory conference at MSRI in 1993, with neither side showing
> any sympathy for the opposing viewpoint nor any inclination to change
> their own position.
>
> 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?
>
> Please email your opinion and any supporting rationale to me and I will
> collate them and try to distill the reasons down to a small set. If you
> switched sides (in either direction) at some time in the past that would
> also be helpful information. I will preserve the anonymity of opinions
> sent privately to me; if you want them publicly known you will need to
> cc this list yourself, at your discretion.
>
> Vaughan Pratt
>
>
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