back September, 18 - Boris M. Schein (University of Arkansas)
 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. back

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