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This is an expanded version of a series of four lectures designed to show algebraists how ring theoretical methods can be used to analyze an interesting family of finite Markov chains. The chains happen to be random walks on semigroups, and the analysis is based on a study of the associated semigroup algebras. The paper

We prove that the pseudovariety DS, of all finite monoids each of whose regular D-classes is a subsemigroup, is local. (A pseudovariety (or variety) V is local if any category whose local monoids belong to V divides a member of V). The proof uses the "kernel theorem" of the first two authors together with the description by P. Weil of DS as an iterated "block product". The one-sided analogues of these methods provide wide new classes of local pseudovarieties of completely regular monoids. We conclude, however, with the third author's example of a variety (and a pseudovariety) of completely regular monoids that is not local. A pseudovariety V of monoids is said to be local if every finite category C whose local (or "loop") monoids belong to V divides a monoid in V. Locality has important applications to both the theory of pseudovarieties and to its connection with varieties of formal languages [20, 10]. The first author showed in [4] and with M. B. Szendrei in [7] that many pse...

The notion of hyperdecidability has been introduced as a tool which is particularly suited for granting decidability of semidirect products. It is shown in this paper that the semidirect product of an hyperdecidable pseudovariety with a pseudovariety whose finitely generated free objects are finite and effectively computable is again hyperdecidable. As instances of this result, one obtains, for example, the hyperdecidability of the pseudovarieties of all finite completely simple semigroups and of all finite bands of left groups. 1. Introduction The semidirect product is one of the most studied algebraic constructions in semigroup theory. In the theory of finite semigroups, applications such as the Krohn-Rhodes complexity have led to the study of the semidirect product as an operation on pseudovarieties and prompted the consideration of the decidability (of the membership) problem for pseudovarieties constructed as semidirect products. Although no example has yet been published, it is ...

: We use classical results on the lattice L(B) of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice P s(DA) of subpseudovarieties of DA, --- where DA is the largest pseudovariety of finite semigroups in which all regular semigroups are band semigroups. We bring forward a lattice congruence on P s(DA), whose quotient is isomorphic to L(B), and whose classes are intervals with effectively computable least and greatest members. Also we characterize the pro-identities satisfied by the members of an important family of subpseudovarieties of DA. Finally, letting V k be the pseudovariety generated by the k-generated elements of DA (k 1), we use all our results to compute the position of the congruence class of V k in L(B). Introduction The lattice of pseudovarieties of finite semigroups has been the object of much attention over the past few decades, with motivations drawn not only from universal algebra, but also from theoretical computer scie...

Several algorithmic problems for pseudovarieties and their relationships are studied. This includes the usual membership problem and the computability of pointlike subsets of finite semigroups. Some of these problems afford equivalent formulations involving topological separation properties in free profinite semigroups. Several examples are considered and, as an application, a decidability result for joins is proved. 1. Introduction Perhaps the three most celebrated results relating the theories of formal languages and finite semigroups are: Schutzenberger's characterization of star-free languages as those whose syntactic semigroups are finite and aperiodic [19]; Simon's characterization of piecewise testable languages as those whose syntactic semigroups are finite and J-trivial [20]; and Brzozowski and Simon / McNaughton 's characterization of locally testable languages as those whose syntactic semigroups are finite local semilattices [7, 15]. These results led Eilenberg [9] to ...

. We prove that any maximal group in the free Burnside semigroup defined by the equation x n = x n+m for any n 1 and any m 1 is a free Burnside group satisfying x m = 1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a graph associated to the J -class containing the group. For n = 2 and for every m 2 we present examples with 2m \Gamma 1 generators. Hence, in these cases, we have infinite maximal groups for large enough m. This allows us to prove important properties of Burnside semigroups for the case n = 2, which was almost completely unknown until now. Surprisingly, the case n = 2 presents simultaneously the complexities of the cases n = 1 and n 3: the maximal groups are cyclic of order m for n 3 but they can have more generators and be infinite for n 2; there are exactly 2 jAj J -classes and they are easily characterized for n = 1 but there are infinitely many J -classes and they are difficult to c...

. A pattern equation is a word equation of the form X = A where X is a sequence of variables and A is a sequence of constants. The problem whether X = A has a solution in a free idempotent semigroup (free band) is shown to be NP#complete. 1 Introduction The main topic of interest in several \Thetaelds of computer science is solving equations. Many areas such as logic programming, automated theorem proving and pattern matching exploit solving equations, and uni\Thetacation is a typical example of it. An important role is played also by semantic uni\Thetacation, which is in fact solving word equations in some variety. Makanin (see [10]) shows that the question whether an equation in a free monoid has a solution is decidable. It can be even generalized in the way that existential \Thetarst-order theory of equations over free monoid is decidable including additional regular constraints on the variables [13]. In this paper we focus on a certain subclass of equations which we call pattern ...

. Word equation in a special form X = A, where X is a sequence of variables and A is a sequence of constants, is considered. The problem whether X = A has a solution over a free monoid (PATTERN-EQUATION problem) is shown to be NP#complete. It is also shown that disjunction of a special type equation systems and conjunction of the general ones can be eliminated. Finally, the case of stuttering equations where the word identity is read modulo x 2 = x is mentioned. 1 Introduction In computer science many natural problems lead to solving equations. It is the main topic in several \Thetaelds such as logic programming and automated theorem proving where especially uni\Thetacation plays a very important role. A number of problems also exploit semantic uni\Thetacation, which is in fact solving word equations in some variety. A very famous result by Makanin (see [10]) shows that the question whether an equation over a free monoid has a solution is decidable. It can be even generalized in the...

Word equation in a special form X = A, where X is a sequence of variables and A is a sequence of constants, is considered. The problem whether X = A has a solution over a free monoid (Pattern-Equation problem) is shown to be NP-complete. It is also shown that disjunction of a special type equation systems and conjunction of the general ones can be eliminated. Finally, the case of stuttering equations where the word identity is read modulo x 2 = x is mentioned.