A variety theorem without complementation

The most important tool for classifying recognizable languages is Eilenberg’s variety theorem [1], which gives a one-to-one correspondence between (pseudo)-varieties of finite semigroups and varieties of recognizable languages. Varieties of recognizable languages are classes of recognizable languages closed under union, intersection, complement, left and right quotients and inverse morphisms. Recall that one passes from a language to a finite semigroup by computing its syntactic semigroup. However, certain interesting families of recognizable languages, which are not varieties of languages, also admit a syntactic characterization. The aim of this paper is to show that such results are not isolated, but are instances of a result as general as Eilenberg’s theorem. On the language side, we consider positive varieties of languages, which have the same properties as varieties of languages except they are not supposed to be closed under complement. On the algebraic side, varieties of finite semigroups are replaced by varieties of finite ordered semigroups. Our main result states there is a one-to-one correspondence between positive varieties of languages and varieties of finite ordered semigroups. Due to the lack of space, we shall just give a few examples of this correspondence and defer to future papers the detailed study of our new types of varieties. For instance, P. Weil and the author have shown that the theorems of Birkhoff and Reiterman can be extended to ordered semigroups by replacing equations by inequations. The proof of the main result is of course inspired by the proof of Eilenberg’s theorem, although there are some subtle adjustments to do. The basic definitions and properties of ordered semigroups are presented in section 2. Recognizable sets are introduced in section 3, but our definition extends the standard one since we are dealing with ordered semigroups. The notion of syntactic ordered semigroup is defined in section 4. The main result is