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

This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture --- also verified by Ash --- it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ash's theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH 1 H 2 \Delta \Delta \Delta Hn , where each H i is a finitely generated subgroup of G. This significantly extends classical results by M. Hall. Final...

We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective algorithm to decide whether a given rational language is expressible. Our main result states a similar condition for the "restricted" temporal logic (RTL), obtained by discarding the "until" operator. A formal language is RTL-expressible if and only if its syntactic semigroup is finite and satisfies a certain simple algebraic condition. This leads

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This paper is a contribution to the algebraic theory of recognizable languages. The main topic of this paper is the polynomial closure, an operation that mixes together the operations of union and concatenation. Formally, the polynomial closure of a class of languages L of A ∗ is the set of languages

This paper is an attempt to share with a larger audience some modern developments in the theory of finite automata. It is written for the mathematician who has a background in semigroup theory but knows next to nothing on automata and languages. No proofs are given, but the main results are illustrated by several examples and counterexamples

We give a general description of the free profinite semigroups over a semidirect product of pseudovarieties. More precisely,\Omega A (V W) is described as a closed subsemigroup of a profinite semidirect product of the form\Omega \Omega AW\ThetaA V \Omega AW. As a particular case, the free profinite semigroup over J 1 V is described in terms of the geometry of the Cayley graph of the free profinite semigroup over V (here J 1 is the pseudovariety of semilattice monoids). Applications are given to the calculations of the free profinite semigroup over J 1 Nil and of the free profinite monoid over J 1 G (where Nil is the pseudovariety of finite nilpotent semigroups and G is the pseudovariety of finite groups). The latter free profinite monoid is compared with the free profinite inverse monoid, which is also calculated here.

this paper is to present other semirings that occur in theoretical computer science. These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min; +)-semirings

We use algebraic techniques to obtain superlinear lower bounds on the size of boundedwidth branching programs to solve a number of problems. In particular, we show that any bounded-width branching program computing a nonconstant threshold function has length \Omega\Gamma n log log n); improving on the previous lower bounds known to apply to all such threshold functions. We also show that any program over a finite solvable monoid computing products in a nonsolvable group has length\Omega\Gamma n log log n): This result is a step toward proving the conjecture that the circuit complexity class ACC 0 is properly contained in NC 1 : A preliminary version of this paper appeared in the Proceedings of the 1991 Structure in Complexity Theory Symposium. 1. The Main Results In this paper we describe a general algebraic technique for obtaining superlinear lower bounds on the length of bounded-width branching programs to solve certain problems. Our method is based on the interpretation, ...

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