Infinite-dimensional triangularizable algebras

Let {\mathrm{End}_{k}(V)} denote the ring of all linear transformations of an arbitrary k-vector space V over a field k. We define {X\subseteq\mathrm{End}_{k}(V)} to be triangularizable if V has a well-ordered basis such that X sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that an arbitrary subset of {\mathrm{End}_{k}(V)} is strictly triangularizable (defined in the obvious way) if and only if it is topologically nilpotent. This generalizes the theorem of Levitzki that every nilpotent semigroup of matrices is triangularizable. We also give a description of the triangularizable subalgebras of {\mathrm{End}_{k}(V)} , which generalizes a theorem of McCoy classifying triangularizable algebras of matrices over algebraically closed fields.

The subpower membership problem for semigroups

Fix a finite semigroup [Formula: see text] and let [Formula: see text] be tuples in a direct power [Formula: see text]. The subpower membership problem (SMP) asks whether [Formula: see text] can be generated by [Formula: see text]. If [Formula: see text] is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in [Formula: see text]. For semigroups this problem always lies in PSPACE. We show that the [Formula: see text] for a full transformation semigroup on [Formula: see text] or more letters is actually PSPACE-complete, while on [Formula: see text] letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup [Formula: see text] embeds into a direct product of a Clifford semigroup and a nilpotent semigroup, then [Formula: see text] is in P; otherwise it is NP-complete.

POINTLIKE SETS, HYPERDECIDABILITY AND THE IDENTITY PROBLEM FOR FINITE SEMIGROUPS

In this paper, we give a relationship between the identity problem and the problem of deciding whether certain subsets of nilpotent semigroups are pointlike. We then use this to give an example of a pseudovariety which has a decidable membership problem, but for which one cannot decide pointlike sets. Then, by modifying the equations, we show that no graph is fundamentally hyperdecidable by constructing, for each graph, a labeling over a nilpotent semigroup for which we cannot decide inevitability with respect to the pseudovariety defined by these equations.

On a class of Noetherian algebras

A class of Noetherian semigroup algebrasK[S]is described. In particular, we show that, for any submonoidSof the semigroupMnof all monomialn × nmatrices over a polycyclic-by-finite groupG, K[S]is right Noetherian if and only ifSsatisfies the ascending chain condition on right ideals. This is then used to prove that every prime homomorphic image of a semigroup algebra of a finitely generated Malcev nilpotent semigroupSsatisfying the ascending chain condition on right ideals is left and right Noetherian.

THE BIDETERMINISTIC CONCATENATION PRODUCT

This paper is devoted to the study of the bideterministic concatenation product, a variant of the concatenation product. We give an algebraic characterization of the varieties of languages closed under this product. More precisely, let V be a variety of monoids, [Formula: see text] the corresponding variety of languages and [Formula: see text] the smallest variety containing [Formula: see text] and the bideterministic products of two languages of [Formula: see text]. We give an algebraic description of the variety of monoids [Formula: see text] corresponding to [Formula: see text]. For instance, we compute [Formula: see text] when V is one of the following varieties: the variety of idempotent and commutative monoids, the variety of monoids which are semilattices of groups of a given variety of groups, the variety of ℛ-trivial and idemptotent monoids. In particular, we show that the smallest variety of languages closed under bideterministic product and containing the language {1}, corresponds to the variety of [Formula: see text]-trivial monoids with commuting idempotents. Similar results were known for the other variants of the concatenation product, but the corresponding algebraic operations on varieties of monoids were based on variants of the semidirect product and of the Malcev product. Here the operation [Formula: see text] makes use of a construction which associates to any finite monoid M an expansion [Formula: see text] with the following properties: (1) M is a quotient of [Formula: see text] (2) the morphism [Formula: see text] induces an isomorphism between the submonoids of [Formula: see text] and of M generated by the regular elements and (3) the inverse image under π of an idempotent of M is a 2-nilpotent semigroup.