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

If $ \Phi$ is a semigroup of functions (= partial transformations) in a set $ A$ then $ \Phi$ is (partially) ordered in a natural way. If $ f,g\in \Phi$ then $ f\le g$ precisely when $ f\subseteq g$, that is, for every $ a\in A$, if $ f(a)$ is defined then $ g(a)$ is defined and $ g(a)=f(a)$.

Every (abstract) semigroup $ S$ is isomorphic to various semigroups $ \Phi$ of functions on various sets $ A$. Under this isomorphism, $ S$ is ordered by an order relation isomorphic to $ \le$ relation on $ \Phi$. These order relations on $ S$ are called fundamental. Thus, fundamental orders of $ S$ are an important subclass of all possible order relations on $ S$. They reflect certain specific properties of $ S$ 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 $ A$. If $ \rho$ is such a relation (that is, $ \rho\subseteq A\times A$) and $ (a,b)\in \rho$ for some $ a,b\in A$, then $ b$ is an ``image" of $ a$ with respect to $ \rho$. Thus $ \rho$ is a many-valued function, each $ a$ may have many (or no) images. Or we may consider ordinary functions (each $ a\in A$ has at most one image under $ \rho$). 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 $ S$.

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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