If is a semigroup of functions (= partial transformations) in a
set then is (partially) ordered in a natural way. If
then precisely when
, that is,
for every , if is defined then is defined and
.
Every (abstract) semigroup is isomorphic to various semigroups
of functions on various sets . Under this isomorphism,
is ordered by an order relation isomorphic to relation on
. These order relations on are called fundamental.
Thus, fundamental orders of are an important subclass of all
possible order relations on . They reflect certain specific
properties of when this semigroup is viewed as a semigroup of
functions.
Very much depends on what we mean by ``functions". The most
general class of functions is binary relations between the elements of
. If is such a relation (that is,
)
and
for some , then is an ``image" of
with respect to . Thus is a manyvalued function,
each may have many (or no) images. Or we may consider ordinary
functions (each has at most one image under ). Or we
may consider onetoone partial transformations only, etc.
The wider is the class of our ``functions", the more fundamental
orderings there may exist on our semigroup .
The speaker will characterize fundamental order relations on
abstract semigroups giving simple necessary and sufficient conditions
for each order relaton to be fundamental.
