Valuation Monoid Research Articles

A characterization of graded pseudo-valuation domains

Let [Formula: see text] be a torsionless grading monoid, [Formula: see text] a [Formula: see text]-graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of [Formula: see text] and [Formula: see text] the group of units of [Formula: see text]. We say that R is a graded pseudo-valuation domain (gr-PVD) if whenever a homogeneous prime ideal P of R contains the product xy of two homogeneous elements of [Formula: see text], then [Formula: see text] or [Formula: see text]. The notion of gr-PVDs was introduced recently by the authors in (M. T. Ahmed, C. Bakkari, N. Mahdou and A. Riffi, Graded pseudo-valuation domains, Comm. Algebra 48 (2020) 4555–4568) as a graded version of pseudo-valuation domains (PVDs). In this paper, we show that R is a gr-PVD if and only if exactly one of the following two conditions holds : (1) (a) [Formula: see text], (b) [Formula: see text] is a pseudo-valuation monoid, and (c) [Formula: see text] for every [Formula: see text] whenever [Formula: see text] is not a unit. (2) (a) [Formula: see text], (b) [Formula: see text] is a valuation monoid, (c) [Formula: see text] for every [Formula: see text] whenever [Formula: see text] is not a unit, and (d) [Formula: see text] is a gr-PVD.

Weighted automata with storage

We consider finite-state automata that are equipped with a storage. Moreover, the transitions are weighted by elements of a unital valuation monoid. A weighted automaton with storage recognizes a weighted language, which is a mapping from input strings to elements of the carrier set of the unital valuation monoid. For the class of weighted languages recognizable by such automata we prove closure properties, a Chomsky-Schützenberger theorem, and a Büchi-Elgot-Trakhtenbrot theorem. In case of idempotent, locally finite, and sequential unital valuation monoids, the recognized weighted languages are step functions.

A Kleene theorem for weighted tree automata over tree valuation monoids

We investigate weighted tree automata over Cauchy tree valuation monoids, a new type of weight structure which includes all commutative semirings and, in addition, average and discounted computations of weights for trees. We define rational tree series over these weight structures, and we prove Kleene's classical theorem for this setting: a tree series over a Cauchy tree valuation monoid is recognizable by a weighted tree automaton if and only if it is rational. The proof works via direct automata-theoretic constructions. Along the way, our results yield a new characterization of weighted tree automata over commutative semirings by rational expressions.

Graded integral domains which are UMT-domains

ABSTRACTLet Γ be a torsionless commutative cancellative monoid, be a Γ-graded integral domain, and H be the set of nonzero homogeneous elements of R. In this paper, we show that if Q is a maximal t-ideal of R with Q∩H = ∅, then RQ is a valuation domain. We then use this result to give simple proofs of the facts that (i) R is a UMT-domain if and only if RQ is a quasi-Prüfer domain for each homogeneous maximal t-ideal Q of R and (ii) R is a PvMD if and only if every nonzero finitely generated homogeneous ideal of R is t-invertible, if and only if RQ is a valuation domain for all homogeneous maximal t-ideals Q of R. Let D[Γ] be the monoid domain of Γ over an integral domain D. We also show that D[Γ] is a UMT-domain if and only if D is a UMT-domain and the integral closure of ΓS is a valuation monoid for all maximal t-ideals S of Γ. Hence, D[Γ] is a PvMD if and only if D is a PvMD and Γ is a Prüfer v-multiplication semigroup.

A Nivat theorem for weighted picture automata and weighted MSO logics

Picture languages have been investigated by several research groups. Here, we define weighted two-dimensional on-line tessellation automata (w2ota) taking weights from a new weight structure called picture valuation monoid. The behavior of this automaton model is a picture series mapping pictures to values. As our first main result, we prove a Nivat theorem for w2ota. It shows a connection of mappings and recognizability of certain simple series. Then, we introduce a weighted MSO logic which models average density of pictures. In the second main result, we show that w2ota and a fragment of our weighted MSO logics are expressively equivalent.

Graded-valuation domains

ABSTRACTLet Γ be a torsionless grading monoid, a Γ-graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of R0, and G0 = Γ∩−Γ the group of units of Γ. We say that R is a graded-valuation domain if either x∈R or x−1∈R for every nonzero homogeneous element x∈RH. In this paper, we show that R is a graded-valuation domain if and only ifΓ is a valuation monoid,Rα = Kx for every 0≠x∈Rα whenever α is not a unit of Γ, and is a graded-valuation domain.Let R = Kγ[X;Γ] be a twisted semigroup ring of Γ over K, C a totally ordered (additive) abelian group, C′ a subgroup of C, μ:K→C′∪{∞} a valuation, φ:Γ→C a function such that C′∪φ(Γ) generates C, and v:R→C∪{∞} the function defined by for every . We show that v is a valuation if and only if μ(γ(a,b))+φ(a+b) = φ(a)+φ(b) for every a,b∈Γ.

A link between multioperator and tree valuation automata and logics

Weighted tree languages over semirings lack the expressive power to model computations like taking the average or the discounting of weights in a straightforward manner. This limitation was overcome by weighted tree automata and logics using (a) tree valuation monoids and (b) multioperator monoids. We compare the expressive power of these two solutions and show that a weighted tree language recognizable (resp. definable) over a tree valuation monoid is also recognizable (resp. definable) using a multioperator monoid. For this, we provide direct, semantic-preserving transformations between the automata models and between the respective logics.

Some supplements to Feferman–Vaught related to the model theory of adeles

We give foundational results for the model theory of AKfin, the ring of finite adeles over a number field, construed as a restricted product of local fields. In contrast to Weispfenning we work in the language of ring theory, and various sortings interpretable therein. In particular we give a systematic treatment of the product valuation and the valuation monoid. Deeper results are given for the adelic version of Krasner's hyperfields, relating them to the Basarab–Kuhlmann formalism.

A Note on the n-Generator Property for Commutative Monoids

Let M be a cancellative, commutative monoid with integral closure $\overline{M}$ . Borrowing from ring theory, we say that M has the n-generator property iff every finitely generated ideal of M can be generated by n elements, and we say M has rank n iff every ideal of M can be generated by n elements. We investigate the integral closure of such monoids. We show, in particular, that if M has the n-generator property, then $\overline{M}$ is a valuation monoid, and if M has rank n, then $\overline{M}$ is a principal ideal monoid.