On the extension of partial orders on semigroups of right quotients.

Abstract

for a partial order Os of S is sufficient in order to extend Os to a partial order 0T of T(l). Beyond this we have proved in [13] that each partial order of S can be extended to a partial order of T, and that one has the following situation: On the one hand, there is a 1-1 correspondence between all partial orders 0T of T and those partial orders OS of S which obey (1). This correspondence is given by mapping each 0T on its restriction OTIS=Os on S and conversely 0T is the smallest extension of OS on T. On the other hand, each partial order OS of S can be extended to such a partial order of S which obeys (1), the smallest of them is unique. In this paper we are going to generalize these results to a semigroup T= QT(S, E) of right quotients of S. For this purpose we give in ?1 a summary about semigroups of right quotients including an outline of a proof of their existence which is essentially given in [10]. It turns out that the first part of the result above is also true for semigroups of right quotients, replacing (1) by a similar two-sided condition (cf. Theorem 2). But the second only holds under supplementary assumptions (cf. Theorem 5), of course including the result above, and we give an example of a partial order on a certain semigroup S, which can not be extended to any semigroup of right quotients of S. Moreover, we obtain some results about the question of extending partial orders of S to different semigroups of right quotients Ti = Q,(S, Ej) (cf. Theorem 3). For concepts and notations not defined in the text we refer to [2] and [5]. The author acknowledges the support of the National Science Foundation (NSF GP-6505).