For V a vector space over a field, or more generally, over a division ring, it is well-known that every x ∈ End(V ) has an inner inverse; that is, that there exists y ∈ End(V ) satisfying xyx = x. We show here that a large class of such x have inner inverses y that satisfy with x an infinite family of additional monoid relations, making the monoid generated by x and y what is known as an invers...

We consider the following sports scheduling problem. Consider 2n teams in a sport league. Each pair of teams must play exactly one match in 2n− 1 days. That is, n games are held simultaneously in a day. We want to make a schedule which has n(2n − 1) games for 2n − 1 days. When we make a schedule, the schedule must satisfy a constraint according to the HAP set, which designates a home game or an...

We present ACME, a tool implementing algebraic techniques to solve decision problems from automata theory. The core generic algorithm takes as input an automaton and computes its stabilization monoid, which is a generalization of its transition monoid. Using the stabilization monoid, one can solve many problems: determine whether a B-automaton (which is a special kind of automata with counters)...

We consider the following sports scheduling problem. Consider 2n teams in a sport league. Each pair of teams must play exactly one match in 2n − 1 days. That is, n games are held simultaneously in a day. We want to make a schedule which has n(2n − 1) games for 2n − 1 days. When we make a schedule, the schedule must satisfy a constraint according to the HAP table, which designates a home game or...

The program-over-monoid model of computation originates with Barrington’s proof that it captures the complexity class NC1. Here we make progress in understanding the subtleties of the model. First, we identify a new tameness condition on a class of monoids that entails a natural characterization of the regular languages recognizable by programs over monoids from the class. Second, we prove that...

Intrinsic complete characterizations of constructive, context-free and regular languages have been formulated by means of configurations of languages. The definition of a semiconfiguration is given here by generalizing the definition of a configuration. By means of semiconfigurations, an intrinsic complete characterization of context-sensitive languages is formulated. 1. Languages and generaliz...

The classical Eilenberg correspondence, based on the concept of the syntactic monoid, relates varieties of regular languages with pseudovarieties of finite monoids. Various modifications of this correspondence appeared, with more general classes of regular languages on one hand and classes of more complex algebraic structures on the other hand. For example, classes of languages need not be clos...

The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. ...

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the maximal syntactic complexity of languages in that subclass, taken as a function of the state complexity of these languages. We study the syntactic complexity of star-free regular languages, that is, languages that can be constructed fro...

A regular language has the zero-one law if its asymptotic density converges to either zero or one. We prove that the class of all zero-one languages is closed under Boolean operations and quotients. Moreover, we prove that a regular language has the zero-one law if and only if its syntactic monoid has a zero element. Our proof gives both algebraic and automata characterisation of the zero-one l...