Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalizing two results on regula...

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 bounds on the syntactic monoid size, proving that an nstate deterministic finite automaton with singleton input alphabet (input alphabet with at least three letters, respectively) induce...

An endomorphism of the free monoid A∗ is invertible if it is injective and extends to an automorphism of the free group generated by A. A simple example: the endomorphism that leaves all generators A invariant except one, say a, which is mapped to ba for some other generator b. We give a monoid presentation for the submonoid generated by all such endomorphisms when a and b are taken arbitrarily...

Let Ω be a finite set and T(Ω) the full transformation monoid on Ω. The rank of t∈T(Ω) is natural number |Ωt|. Given A⊆T(Ω), denote by 〈A〉 semigroup generated A. k fixed such that 2≤k≤|Ω|. In first part this paper we (almost) classify permutation groups G for all transformations t∈T(Ω), every element in St:=〈G,t〉 can written as product eg, where e2=e∈St g∈G. second prove, among other results, i...

In this study, Bruck-Reilly extension of a ternary monoid is defined. Additionally, some results about construction are given which belongs to one the classes semigroups; regular, inverse, orthodox and strongly regular.