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The variety generated by the Brandt semigroup ${bf B}_2$ can be defined within the variety generated by the semigroup ${bf A}_2$ by the single identity $x^2y^2approx y^2x^2$. Edmond Lee asked whether or not the same is true for the monoids ${bf B}_2^1$ and ${bf A}_2^1$. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of ${bf A}_2^1$ that satisfy $x^2y^2approx y^2x^2$ and contain ${bf B}_2^1$. A further consequence is that the variety of ${bf B}_2^1$ cannot be defined within the variety of ${bf A}_2^1$ by any finite system of identities. Continuing downward, we then turn to subvarieties of ${bf B}_2^1$. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity $x^2yapprox yx^2$ and containing the monoid $M({bf z}_infty)$, where ${bf z}_infty$ denotes the infinite limit of the Zimin words ${bf z}_0=x_0$, ${bf z}_{n+1}={bf z}_n x_{n+1}{bf z}_n$.
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Graded modalities have been proposed in recent work on programming languages as a general framework for refining type systems with intensional properties. In particular, continuous endomaps of the discrete time scale, or time warps, can be used to quantify the growth of information in the course of program execution. Time warps form a complete residuated lattice, with the residuals playing an important role in potential programming applications. In this paper, we study the algebraic structure of time warps, and prove that their equational theory is decidable, a necessary condition for their use in real-world compilers. We also describe how our universal-algebraic proof technique lends itself to a constraint-based implementation, establishing a new link between universal algebra and verification technology.
We take a long magical tour in algebraic logic, starting from classical results on neat embeddings due to Henkin, Monk and Tarski, all the way to recent results in algebraic logic using so--called rainbow constructions invented by Hirsch and Hodkinson. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. The paper has a survey character but it contains new approaches to old ones. We aspire to make our survey fairly comprehensive, at least in so far as Tarskian algebraic logic, specifically, the theory of cylindric algebras, is concerned. Other topics, such as abstract algebraic logic, modal logic and the so--called (central) finitizability problem in algebraic logic will be dealt with; the last in some detail. Rainbow constructions are used to solve problems adressing classes of cylindric--like algebras consisting of algebras having a neat embedding property. The hitherto obtained results generalize seminal results of Hirsch and Hodkinson on non--atom canonicity, non--first order definabiity and non--finite axiomatizability, proved for classes of representable cylindric algebras of finite dimension$>2$. We show that such results remain valid for cylindric algebras possesing relativized {it clique guarded} representations that are {it only locally} well behaved. The paper is written in a way that makes it accessible to non--specialists curious about the state of the art in Tarskian algebraic logic. Reaching the boundaries of current research, the paper also aspires to be informative to the practitioner, and even more, stimulates her/him to carry on further research in main stream algebraic logic.
This paper explores how a pluralist view can arise in a natural way out of the day-to-day practice of modern set theory. By contrast, the widely accepted orthodox view is that there is an ultimate universe of sets $V$, and it is in this universe that mathematics takes place. From this view, the purpose of set theory is learning the truth about $V$. It has become apparent, however, that the phenomenon of independence - those questions left unresolved by the axioms - holds a central place in the investigation. This paper introduces the notion of independence, explores the primary tool (soundness) for establishing independence results, and shows how a plurality of models arises through the investigation of this phenomenon. Building on a familiar example from Euclidean geometry, a template for independence proofs is established. Applying this template in the domain of set theory leads to a consideration of forcing, the tool par excellence for constructing universes of sets. Fifty years of forcing has resulted in a profusion of universes exhibiting a wide variety of characteristics - a multiverse of set theories. Direct study of this multiverse presents technical challenges due to its second-order nature. Nonetheless, there are certain nice local neighborhoods of the multiverse that are amenable to first-order analysis, and emph{set-theoretic geology} studies just such a neighborhood, the collection of grounds of a given universe $V$ of set theory. I will explore some of the properties of this collection, touching on major concepts, open questions, and recent developments.
This is a collection of statements gathered on the occasion of the Quantum Physics of Nature meeting in Vienna.
We revisit the geometry of involutions in groups of finite Morley rank. Our approach unifies and generalises numerous results, both old and recent, that have exploited this geometry; though in fact, we prove much more. We also conjecture that this path leads to a new identification theorem for $operatorname{PGL}_2(mathbb{K})$.
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