Aperiodic semigroup

Abstract: The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of the minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having $n$ left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with $n$ states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For $n \ge 4$ there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that $2^n-1$ is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.

Abstract: We consider implicit signatures over finite semigroups dete rmined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors withi n the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profin ite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.

Abstract: Profinite semigroups provide powerful tools to understand properties of classes of regular languages. Until very recently however, little was known on the structure of "large" relatively free profinite semi- groups. In this paper, we present new results obtained for the class of all finite aperiodic (that is, group-free) semigroups. Given a finite al- phabet X, we focus on the following problems: (1) the word problem for ω-terms on X evaluated on the free pro-aperiodic semigroup, and (2) the computation of closures of regular languages in the ω-subsemigroup of the free pro-aperiodic semigroup generated by X.

Abstract: Aword w is called synchronizing (recurrent, reset, directable) word of deterministic finite automaton (DFA) if w brings all states of the automaton to an unique state. Cerny conjectured in 1964 that every n- state synchronizable automaton possesses a synchronizing word of length at most (n - 1)2. The problem is still open. It will be proved that the minimal length of synchronizing word is not greater than (n - 1)2/2 for every n-state (n > 2) synchronizable DFA with transition monoid having only trivial subgroups (such automata are called aperiodic). This important class of DFA accepting precisely star-free languages was involved and studied by Schutzenberger. So for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true. Some properties of an arbitrary synchronizable DFA and its transition semigroup were established. http://www.cs.biu.ac.il/~trakht/syn.html