The family ${\mathcal{R}} X^*$ of regular subsets of the free monoid $X^*$ generated by a finite set X is the standard example of a ${}^*$ -continuous Kleene algebra. Likewise, the family ${\mathcal{C}} X^*$ of context-free subsets of $X^*$ is the standard example of a $\mu$ -continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator $\mu$ . For arbitrary monoids M, ${\mathcal{C}} M$ is the closure of ${\mathcal{R}}M$ as a $\mu$ -continuous Chomsky algebra, more briefly, the fixed-point closure of ${\mathcal{R}} M$ . We provide an algebraic representation of ${\mathcal{C}} M$ in a suitable product of ${\mathcal{R}} M$ with $C_2'$ , a quotient of the regular sets over an alphabet $\Delta_2$ of two pairs of bracket symbols. Namely, ${\mathcal{C}}M$ is isomorphic to the centralizer of $C_2'$ in the product of ${\mathcal{R}} M$ with $C_2'$ , i.e. the set of those elements that commute with all elements of $C_2'$ . This generalizes a well-known result of Chomsky and Schützenberger (1963, Computer Programming and Formal Systems, 118–161) and admits us to denote all context-free languages over finite sets $X\subseteq M$ by regular expressions over $X\cup\Delta_2$ interpreted in the product of ${\mathcal{R}} M$ and $C_2'$ . More generally, for any ${}^*$ -continuous Kleene algebra K the fixed-point closure of K can be represented algebraically as the centralizer of $C_2'$ in the product of K with $C_2'$ .

This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus.

We show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

We tackle the problem of studying which kind of functions can occur as complexity functions of formal languages of a certain type. We prove that an important narrow subclass of rational languages contains languages of polynomial complexity of any integer degree over any non-trivial alphabet.

We discuss some known and introduce some new hierarchies and reducibilities on regular languages, with the emphasis on the quantifier-alternation and difference hierarchies of the quasi-aperiodic languages. The non-collapse of these hierarchies and decidability of some levels are established. Complete sets in the levels of the hierarchies under the polylogtime and some quantifier-free reducibilities are found. Some facts about the corresponding degree structures are established. As an application, we characterize the regular languages whose balanced leaf-language classes are contained in the polynomial hierarchy. For any discussed reducibility we try to give motivations and open questions, in a hope to convince the reader that the study of these reducibilities is interesting for automata theory and computational complexity.

In our main result, we establish a formal connection between Lindström quantifiers with respect to regular languages and the double semidirect product of finite monoids with a distinguished set of generators. We use this correspondence to characterize the expressive power of Lindström quantifiers associated with a class of regular languages.

A deterministic automaton recognizing a givenω-regular languageis constructed from an ω-regular expressionwith the help of derivatives.The construction is related to Safra's algorithm, in about the same way as the classicalderivative method is related to the subset construction.