The intersection of free submonoids of a free monoid is free

Several necessary and sufficient conditions exist for a submonoid of a free monoid to be free. For a few, see [1] through [6]. This short note uses one of these conditions, due to Schützenberger (see [2], p. 119, or [4], Theorem 1.4) to establish that the intersection of free submonoids of a free monoid M is again free. Schützenberger's condition is also proven in this note.     

Stably free modules over virtually free groups

Let F _m be the free group on m generators, and let G be a finite nilpotent group of non square-free order; we show that for each m ≥ 2 the integral group ring Z[G × F _m ] has infinitely many stably free modules of rank 1.     

On optimal factorization of free semigroups into free subsemigroups

Optimal factorizations of a subsemigroup into very pure and weakly prefix subsemigroups are considered. It is shown that the intersection of the family of very pure (weakly prefix) subsemigroups containing a given subset A of X⁺ is itself a very pure (weakly prefix) subsemigroup and procedures for constructing the basis of such minimal subsemigroups are provided.     

Measure free martingales

We give a necessary and sufficient condition on a sequence of functions on a set Ω under which there is a measure on Ω which renders the given sequence of functions a martingale. Further such a measure is unique if we impose a natural maximum entropy condition on the conditional probabilities.     

Some open problems concerning relatively free and free profinite semigroups

Besides constituting an important and often crucial tool in studying semigroup varieties and pseudovarieties, relatively free and free profinite semigroups have proved to be of interest from many other natural standpoints. We present three groups of new problems concerning certain relatively free and free profinite semigroups and give background and motivation for them. The problems deal with the first-order theories of free profinite semigroups, the structure of free Burnside semigroups with commuting idempotents and of free Burnside inverse semigroups, and the representation of finitely generated free bands by order preserving transformations of a finite chain. April 19, 2001     

Subgroups of free idempotent generated semigroups need not be free

We study the maximal subgroups of free idempotent generated semigroups on a biordered set by topological methods. These subgroups are realized as the fundamental groups of a number of 2-complexes naturally associated to the biorder structure of the set of idempotents. We use this to construct the first example of a free idempotent generated semigroup containing a non-free subgroup.     

Almost free algebras

We deal with the class of cardinalsk for which a given variety has an almost free, not free, even not essentially free algebra. The main result is that when enough reflection holds, this behaviour is determined by min{n:n=ω or the variety fail a condition CP _n }. Research supported by NSERC grant #9848. Prof. Mekler died on June 10, 1992. Publication #366. Research supported by the BSF.     

Recubulating free groups

We show that for each finitely generated infinite index subgroup H of a free group F, there is a proper cocompact action of F on a CAT(0) cube complex with a single orbit of hyperplane stabilized by conjugates of H.     

Relatively free bands

The operators h_n and i _n and their duals hmacr;_n and imacr;_n defined on the free semigroup X ⁺ for a nonempty set X by Gerhard and Petrich are proved here to be homomorphisms of onto certain subsets of X ⁺ with a suitable multiplication. From the work of the authors mentioned, these operators induce fully invariant congruences on X ⁺ corresponding to join irreducible varieties of bands if X is countably infinite. New operators on X ⁺ are defined by means of these operators which give homomorphisms in an analogous way and induce fully invariant congruences on X ⁺corresponding to all varieties of bands except for the variety of all bands, and some varieties of normal bands. The former of these was investigated by the mentioned authors and the latter must be treated differently. By means of the above operators we are able to characterize all cases of relatively free bands.     

Torsion free modules

Sunto Si studia un funtore contravariante dalla categoria di tutti gli anelli associativi con identità (con opportuni omomorfismi) nella categoria dei reticoli Booleani completi (equivalentemente, degli anelli Booleani completi). Per ogni anello R il reticolo (R) è l'insieme delle classi di equivalenza [A] modulo un'opportuna equivalenza, ove A varia nella classe degli R-moduli destri non singolari.

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The author is grateful to his colleagueLaszlo Fuchs for his advice and encouragement. The author thanks the referee for a careful reading of the paper, and his useful comments.     

On free p-superalgebras

In this article we consider several aspects of algebraic combinatorics and combinatorial algebra over fields of prime characteristics. P-super-Radford theorem gives the structure of the free associative algebra over a field of prime characteristic with the new multiplication given by the super shuffle product, we show that this algebra is isomorphic to the reduced free super commutative algebra on s-regular words. We prove the elimination theorem for free partially commutative color Lie p-superalgebras and obtain a Schreier type formula for free Lie p-superalgebras using formal power series techniques.