New upper bounds for the greatest number of proper colorings of a (V,E)-graph
Journal of Graph Theory
Varieties Of Formal Languages
Revised Papers from the Second International Workshop on Implementing Automata
WIA '97 Revised Papers from the Second International Workshop on Implementing Automata
State complexity and the monoid of transformations of a finite set
CIAA'04 Proceedings of the 9th international conference on Implementation and Application of Automata
On the size of inverse semigroups given by generators
Theoretical Computer Science
Syntactic complexity of ideal and closed languages
DLT'11 Proceedings of the 15th international conference on Developments in language theory
Syntactic complexity of Prefix-, Suffix-, and Bifix-free regular languages
DCFS'11 Proceedings of the 13th international conference on Descriptional complexity of formal systems
Syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages
Theoretical Computer Science
Syntactic complexities of some classes of star-free languages
DCFS'12 Proceedings of the 14th international conference on Descriptional Complexity of Formal Systems
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We investigate the relationship between regular languages and syntactic monoid size. In particular, we consider the transformation monoids of n-state (minimal) deterministic finite automata. We show tight upper and lower bounds on the syntactic monoid size depending on the number of generators (input alphabet size) used. It turns out, that the two generator case is the most involved one. There we show a lower bound of nn (1 - 2/√n) for the size of the syntactic monoid of a language accepted by an n-state deterministic finite automaton with binary input alphabet. Moreover, we prove that for every prime n ≥ 7, the maximal size semigroup w.r.t. its size among all (transformation) semigroups which can be generated with two generators, is generated by a permutation with two cycles (of appropriate lengths) and a non-bijective mapping merging elements from these two cycles. As a by-product of our investigations we determine the maximal size among all semigroups generated by two transformations, where one is a permutation with a single cycle and the other is a non-bijective mapping.