Centro de Matematica - Universidade do Minho

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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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