Re: algebraic elements

In article <21d1d6f0-7c98-4024-90bd-4e7efa11ca9c@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
=?ISO-8859-1?Q?Mariano_Su=E1rez=2DAlvarez?= <mariano.suarezalvarez@xxxxxxxxx> wrote:
On Nov 23, 8:34=A0pm, Timothy Murphy <t...@xxxxxxxxxxxx> wrote:

[...]

I wonder if that is normal usage?
I see <http://en.wikipedia.org/wiki/Minimal_element> says:
"In mathematics, especially in order theory, a maximal element of a subse=
t S
of some partially ordered set is an element of S
that is not smaller than any other element in S.
The term minimal element is defined dually."

That is certainly what I have always understood;
minimal is simply the adjectival form of minimum.

But that is *not* what Wikipedia is saying.

Indeed. A maximal element need not be a maximum element, just like a
minimal element need not be a minimal element.

An element is a maximum if it is greater than all other elements; it
is maximal if there is no other element strictly larger than it. The
distinction arises in sets that are not totally ordered (two elements
may be incomparable). Likewise, an element is a minimum if it is
smaller than all other elements, and minimal if there is no element
strictly larger than it.

According to Wikipedia and, in my experience,
to mathematical usage, a minimum element is
minimal but a minimal element is not necessarily
a minimum element (in a discretely ordered set
every element is minimal, yet there is no minimum
element unless the set has exactly one element)

Thus minimal is not the adjectival form of minimum.

Indeed.

I ran into a similar problem recently when I suggested to a student
that it would be more grammatical in a proof to write "due to the
primality of P", only to be advised by a ring theorist that there is
such a thing as "primal ideal", which is different from "prime ideal".

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.