The dual symmetric inverse monoid I ∗ n is the inverse monoid of all isomorphisms between quotients of an n-set. We give a monoid presentation of I ∗ n and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.

A growing body of research into the expressive power of logics on trees has employed algebraic methods—especially the syntactic forest algebra, a generalization of the syntactic monoid of regular languages. Here we enlarge the mathematical foundations of this study by extending Tilson’s theory of the algebra of finite categories, and in particular, the Derived Category Theorem, to the setting o...

A language L over an alphabet Σ is prefix-convex if, for any words x, y, z ∈ Σ, whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefixclosed, and prefix-free languages. We study complexity properties of prefix-convex regular languages. In particular, we find the quotient/state complexity of boolean operations, product (concatenation), star, and reversal,...

Based on different concepts to obtain a finer notion of language recognition via finite monoids we develop an algebraic structure called typed monoid. This leads to an algebraic description of regular and non regular languages. We obtain for each language a unique minimal recognizing typed monoid, the typed syntactic monoid. We prove an Eilenberg-like theorem for varieties of typed monoids as w...

We give a transparent characterization, by means of a certain syntactic semigroup, of regular languages possessing the finite power property. Then we use this characterization to obtain a short elementary proof for the uniform decidability of the finite power property for rational languages in all monoids defined by a confluent regular system of deletion rules. This result in particular covers ...

The dual symmetric inverse monoid I ∗ n is the inverse monoid of all isomorphisms between quotients of an n-set. We give a monoid presentation of I ∗ n and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.

A regular language L over an alphabet A is called piecewise testable if it is a finite boolean combination of languages of the form Aa1A a2A ∗ . . . Aa`A ∗, where a1, . . . , a` ∈ A, ` ≥ 0. An effective characterization of piecewise testable languages was given in 1972 by Simon who proved that a language L is piecewise testable if and only if its syntactic monoid is J -trivial. Nowadays there e...

By a generalized Tannaka-Krein reconstruction we associate to the admissible representions of the category O of a Kac-Moody algebra, and its category of admissible duals a monoid with a coordinate ring. The Kac-Moody group is the Zariski open dense unit group of this monoid. The restriction of the coordinate ring to the Kac-Moody group is the algebra of strongly regular functions introduced by ...

This paper shows that a finitely presented monoid with linear Dehn function need not have a regular cross-section, strengthening the previously-known result that such a monoid need not be presented by a finite complete string rewriting system, and contrasting the fact that finitely presented groups with linear Dehn function always have regular cross-sections.