A pomonoid S is a monoid equipped with a partial order that is compatible with the binary operation. In the same way that M -acts over a monoid M correspond to the representation of M by transformations of sets, S-posets correspond to the representation of a pomonoid S by order preserving transformations of posets. Following standard terminology from the theories of R-modules over a unital ring...

In this paper, we study the languages of labeled finite birooted trees: Munn’s birooted trees extended with vertex labeling. We define a notion of finite state birooted tree automata that is shown to capture the class of languages that are upward closed w.r.t. the natural order and definable in Monadic Second Order Logic. Then, relying on the inverse monoid structure of labeled birooted trees, ...

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...

An integer partition λ ` n corresponds, via its Ferrers diagram, to an artinian monomial ideal I ⊂ C[x, y] with dimC C[x, y]/I = n. If λ corresponds to an integrally closed ideal we call it concave . We study generating functions for the number of concave partitions, unrestricted or with at most r parts. 1. concave partitions By an integer partition λ = (λ1, λ2, λ3, . . . ) we mean a weakly dec...

The study of languages of labelled birooted trees, that is, elements of the free inverse monoid enriched by a vertex labelling, has led to the notion of quasi-recognisability. It generalises the usual notion of recognisability by replacing homomorphisms by certain prehomomorphism into finite ordered monoids, called adequate, that only preserve some products: the so-called disjoint ones. In this...

Let (S,≤) be a strictly totally ordered monoid, R be a commutative ring and M be an R-module. We show the following results: (1) If (S,≤) satisfies the condition that 0 ≤ s for all s ∈ S, then the module [[MS,≤]] of generalized power series is a semi Hopfian [[RS,≤]]-module if and only if M is a semi Hopfian R-module; (2) If (S,≤) is artinian, then the generalized inverse polynomial module [MS,...

Classical Buchberger theory is generalized to a new family of rings. The family includes all subalgebras of the polynomial algebra in one variable. Some subalgebras of polynomial algebras in several variables are included. The new rings are integral domains and have a number of other properties in common with polynomial rings. The rings sit in a field F in appropriate position relative to a val...