We solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a suffix-free language with n left quotients (that is, with state complexity n) is at most (n − 1) + n − 2 for n > 6. Since this bound is known to be reachable, this settles the problem. We also reduce the alphabet of the witness languages reaching this bound to five letters ins...

Let S be a finite non-commutative semigroup. The commuting graph of S, denoted G(S), is the graph whose vertices are the non-central elements of S and whose edges are the sets {a, b} of vertices such that a 6= b and ab = ba. Denote by T (X) the semigroup of full transformations on a finite set X. Let J be any ideal of T (X) such that J is different from the ideal of constant transformations on ...

We show that for every homomorphism from A to a finite semigroup S there exists a factorization forest of height at most 3 |S|−1. Furthermore, we show that for every non-trivial group, this bound is tight. For aperiodic semigroups, we give an improved upper bound of 2 |S| and we show that for every n ≥ 2 there exists an aperiodic semigroup S with n elements which reaches this bound.

To a strongly continuous semigroup ? T(t) t0 on a Banach space X we will associate semigroups ? T n (t) t0 on new Banach spaces X n for each n 2 Z. This construction is inspired by the classical Sobolev spaces and, due to its simplicity, of great help in understanding abstract and concrete semigroups. We start with a strongly continuous semigroup ? T(t) t0 on a Banach space X for which we assum...

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I n λ of the rank 6 n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. W...