444 research outputs found

    Semilattices of finitely generated ideals of exchange rings with finite stable rank

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    We find a distributive (v, 0, 1)-semilattice S of size aleph_1 aleph\_1 that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: - There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotent-generated two-sided ideals is isomorphic to S. - There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to S. These results are established by constructing an infinitary statement, denoted here by URPsr, that holds in the maximal semilattice quotient of every Riesz monoid endowed with an order-unit of finite stable rank, but not in the semilattice S

    On positive commutative tomonoids

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    We discuss totally ordered monoids (or tomonoids, for short) that are commutative, positive, and finitely generated. Tomonoids of this kind correspond to certain preorders on free commutative monoids. In analogy to positive cones of totally ordered groups, we introduce direction cones to describe the preorders in question and we establish between both notions a Galois connection. In particular, we show that any finitely generated positive commutative tomonoid is the quotient of a tomonoid arising from a direction cone. We furthermore have a closer look at formally integral tomonoids and at nilpotent tomonoids. In the latter case, we modify our approach in order to obtain a description that is based on purely finitary means.I 1923-N25(VLID)342647

    Totally ordered commutative monoids

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    A totally ordered monoid - or tomonoid, for short - is a commutative semigroup with identity S equipped with a total order ≤s that is translation invariant, i.e., that satisfies: ∀x, y, z ∈, x ≤s y ⇒ x + z ≤s y + z. We call a tomonoid that is a quotient of some totally ordered free commutative monoid formally integral. Our most significant results concern characterizations of this condition by means of constructions in the lattice Zn that are reminiscent of the geometric interpretation of the Buchberger algorithm that occurs in integer programming. In particular, we show that every two-generator tomonoid is formally integral. In addition, we give several (new) examples of tomonoids that are not formally integral, we present results on the structure of nil tomonoids and we show how a valuation-theoretic construction due to Hion reveals relationships between formally integral tomonoids and ordered commutative rings satisfying a condition introduced by Henriksen and Isbell

    How to advance general game playing artificial intelligence by player modelling

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    7 pagesGeneral game playing artificial intelligence has recently seen important advances due to the various techniques known as 'deep learning'. However the advances conceal equally important limitations in their reliance on: massive data sets; fortuitously constructed problems; and absence of any human-level complexity, including other human opponents. On the other hand, deep learning systems which do beat human champions, such as in Go, do not generalise well. The power of deep learning simultaneously exposes its weakness. Given that deep learning is mostly clever reconfigurations of well-established methods, moving beyond the state of art calls for forward-thinking visionary solutions, not just more of the same. I present the argument that general game playing artificial intelligence will require a generalised player model. This is because games are inherently human artefacts which therefore, as a class of problems, contain cases which require a human-style problem solving approach. I relate this argument to the performance of state of art general game playing agents. I then describe a concept for a formal category theoretic basis to a generalised player model. This formal model approach integrates my existing 'Behavlets' method for psychologically-derived player modelling: Cowley, B., Charles, D. (2016). Behavlets: a Method for Practical Player Modelling using Psychology-Based Player Traits and Domain Specific Features. User Modeling and User-Adapted Interaction, 26(2), 257-306.Non peer reviewe

    Regular Cost Functions, Part I: Logic and Algebra over Words

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    The theory of regular cost functions is a quantitative extension to the classical notion of regularity. A cost function associates to each input a non-negative integer value (or infinity), as opposed to languages which only associate to each input the two values "inside" and "outside". This theory is a continuation of the works on distance automata and similar models. These models of automata have been successfully used for solving the star-height problem, the finite power property, the finite substitution problem, the relative inclusion star-height problem and the boundedness problem for monadic-second order logic over words. Our notion of regularity can be -- as in the classical theory of regular languages -- equivalently defined in terms of automata, expressions, algebraic recognisability, and by a variant of the monadic second-order logic. These equivalences are strict extensions of the corresponding classical results. The present paper introduces the cost monadic logic, the quantitative extension to the notion of monadic second-order logic we use, and show that some problems of existence of bounds are decidable for this logic. This is achieved by introducing the corresponding algebraic formalism: stabilisation monoids.Comment: 47 page
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