Abstract: During the last decade or so, there has
been considerable interest in the
algebraic properties of
transformations of sets vis-à-vis those of linear
transformations of
a vector space. Many years ago, following some of
Gould?s ideas,
Fountain and Lewin (1992-1993) used ?independence algebras?
to unify
the two areas. But they had to impose some restrictions on their
algebras
in order to explain very specific differences between the two
areas.
Hence, as more differences were found, it became harder to explain
them.
So, nowadays many people continue to work in one (or both) of these
areas
without using algebras of any kind.
Although these differences
are usually subtle, they are substantial. Also,
surprisingly, the
results for vector spaces are often more complete and
more elegant
than the corresponding ones for sets (also the proofs are
completely
different). In this talk, I will discuss one or more results of
this
kind for the following topics.
--skew idempotents, by Blyth
& Almeida Santos (2006), Changphas & Sullivan
(to appear)
--restricted
ranges, by Sommanee & Sanwong (2007), Sullivan (2008)
--BQ-property,
by Kemprasit (2001-2003), Sullivan (2009) |