Models of a K-Rational Identity System

A rational language can be studied from different points of view. For instance, it can be seen as a part of the free monoid or as a set recognized by a finite automaton; this leads to combinatoric or graphic studies. It can also be considered as a series with coefficients in the boolean semiring; this second point of view is richer since it leads to generalize naturally the theory of rational languages by the theory of rational series with coefficients in an arbitrary semiring. But, there is a third and much less studied point of view which consists in considering a rational language as the interpretation of a rational expression, that is to say, of a formal expression (i.e., of a tree) which represents the given language. This notion leads immediately to difficulties because we have lost the unicity of a language representation since several rational expressions can now denote a same language; nevertheless, it brings us to the important notion of rational identity, initiated by Conway [S]. A problem of logical nature occurs then immediately: how can a complete identity system d, i.e., permitting obtainment from d of every identity by a logical deduction process, be constructed? This difficult problem is solved in [ 131 in the usual case. In this paper, we shall specially interest ourselves in an extension of the usual rational expressions theory which consists in introducing expressions with multiplicities in a general semiring K, interpretated in K((A* )). In order to cover all practical cases, we were led to develop two separated theories according as K is Kleene or not. We shall study here particularly the notion of model of a K-rational identity system and its consequences. Indeed, owing to it, we shall show that every complete K-rational identity system over a positive semiring K is necessarily infinite. This result was already known for K= &I’, where a relatively rapid proof of 396 OO22OOOO/92 $5.00