Let R be a commutative Noetherian ring of dimension d and M cancellative torsion-free seminormal monoid. Then: (1) A type R[d,m,n] P projective A[M]-module rank r≥max {2,d+1}. Then the action E(A[M]⊕P) on (A[M]⊕P) is transitive (2) Assume (R,m,K) regular local containing field k such that either char k=0 or k=p tr-deg K∕

In 1970, R. S. Cohen and Janusz A. Brzozowski introduced a hierarchy of star-free languages called the dot-depth hierarchy. This hierarchy and its generalisations, together with the problems attached to them, had a long-lasting influence on the development of automata theory. This survey article reports on the numerous results and conjectures attached to this hierarchy. This paper is a follow-u...

We consider multiplicative monoids of the positive integers defined by a single congruence. If a and b are positive integers such that a≤ b and a2≡ a mod b, then such a monoid (known as an arithmetic congruence monoid or an ACM) can be described as Ma,b = (a+ bN0)∪{1}. In lectures on elementary number theory, Hilbert demonstrated to students the utility of the proof of the Fundamental Theorem o...

In this paper we investigate several questions related to syntactic congruences and to minimal automata associated with ω-languages. In particular we investigate relationships between the so-called simple (because it is a simple translation from the usual definition in the case of finitary languages) syntactic congruence and its infinitary refinement (the iteration congruence) investigated by A...

Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids that generate monoid varieties with continuum many subvarieties; these include any finite inherently ...

Proper extensions that are “injective on L-related idempotents” of R-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a λ-semidirect product of a left regular band by an R-unipotent or by a glrac semigroup, respectivel...

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic ...

The main result of this paper is as follows: For any commutative regular (actually even K2-regular) ring R and any finitely generated intermediate monoid ZP+ c A4 c Q’+ (for some natural r) the following conditions are equivalent: (a) A4 zE:, (b) R[M] is K,-regular, (c) M is seminormal and SK,(R) = SK,(R[M]) (i.e. the natural homomorphism SK,(R) -SK,(R[M]) is an isomorphism), and, if in additio...

Let Ω be a finite set and T(Ω) the full transformation monoid on Ω. The rank of t∈T(Ω) is natural number |Ωt|. Given A⊆T(Ω), denote by 〈A〉 semigroup generated A. k fixed such that 2≤k≤|Ω|. In first part this paper we (almost) classify permutation groups G for all transformations t∈T(Ω), every element in St:=〈G,t〉 can written as product eg, where e2=e∈St g∈G. second prove, among other results, i...

We define and study the Picard group of a monoid scheme and the class group of a normal monoid scheme. To do so, we develop some ideal theory for (pointed abelian) noetherian monoids, including primary decomposition and discrete valuations. The normalization of a monoid turns out to be a monoid scheme, but not always a monoid.