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Jackson and Lee proved that certain six-element monoid generates a hereditarily finitely based variety E 1 whose lattice of subvarieties contains an infinite ascending chain. We identify syntactic monoids which generate finitely generated subvarieties of E 1 and show that one of these finite monoids together with certain seven-element monoid generates a new limit variety. [ABSTRACT FROM AUTHOR]

Beta-thalassemia is an autosomal recessive blood disorder characterized by abnormalities in the synthesis of ß globin. Together with a globin, it is a subunit of globin protein, called hemoglobin, located inside our red blood cells to deliver oxygen from the lungs to all of the tissues throughout our body. Thereby, individuals with ß-thalassemia will often feel limp due to a lack of oxygen dissolved in their blood. In this paper, a finite state automaton to detect and classify ß globin gene mutations using its DNA sequence is constructed. Finite state automata have a close connection to an algebraic structure, that is, a monoid. Together with the theory of the syntactic monoid, we present a methodology to minimize the number of the internal states of an automaton to have minimal state automata. Therefore, a minimal state automaton can be constructed to detect ß globin gene mutation causing the ß-thalassemia disease. We have developed a MATLAB program to conduct the appropriate simulations. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the group of a rational maximal bifix code Z to be isomorphic with the F-group of Z n F,when F is recurrent and Z n F is rational. The casewhere F is uniformly recurrent, which is known to imply the finiteness of Z n F, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of F. [ABSTRACT FROM AUTHOR]

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic ordered monoids of Pin (D = posets), the syntactic semirings of Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is a commutative variety of algebras or ordered algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in the case where the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids. [ABSTRACT FROM AUTHOR]

We study the structure of the language of binary overlap-free words, which is one of the classical objects in combinatorics of words and formal languages. In a natural way, this study requires a solution to the word problem for the syntactic monoid of the language. In this paper we present an algorithm that efficiently solves this word problem. Namely, the time complexity of the algorithm is linear in the total length of compared words. [ABSTRACT FROM AUTHOR]

The classes of languages which are Boolean combinations of languages of the form $$ A^*\,a_1 A^*\,a_2 A^*\, \ldots A^*\,a_\ell A^*,\,{\rm{where}}\,a_1 ,\,a_2 , \ldots ,\,a_\ell \, \in \,A,\,\ell \, \le \,k, $$ for a fixed k ≥ 0, form a natural hierarchy within piecewise testable languages and have been studied by Simon, Blanchet-Sadri, Volkov and others. The main issues were the existence of finite bases of identities for the corresponding pseudovarieties of monoids and monoids generating these pseudovarieties. Here we deal with similar questions concerning the finite unions and positive Boolean combinations of the languages of the form above. In the first case the corresponding pseudovarieties are given by a single identity, in the second case there are finite bases for k equals to 1 and 2 and there is no finite basis for k ≥ 4 (the case k = 3 remains open). All the pseudovarieties are generated by a single algebraic structure. [ABSTRACT FROM AUTHOR]