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 many-valued 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 one-to-one 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.
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