The regular algebra of a quiver

Let $K$ be a fixed field. We attach to each column-finite quiver $E$ a von Neumann regular $K$-algebra $Q(E)$ in a functorial way. The algebra $Q(E)$ is a universal localization of the usual path algebra $P(E)$ associated with $E$. The monoid of isomorphism classes of finitely generated projective right $Q(E)$-modules is explicitly computed.Comment: 29 page

Code, space and everyday life

In this paper we examine the role of code (software) in the spatial formation of collective life. Taking the view that human life and coded technology are folded into one another, we theorise space as ontogenesis. Space, we posit, is constantly being bought into being through a process of transduction – the constant making anew of a domain in reiterative and transformative practices - as an incomplete solution to a relational problem. The relational problem we examine is the ongoing encounter between individuals and environment where the solution, to a greater or lesser extent, is code. Code, we posit, is diversely embedded in collectives as coded objects, coded infrastructure, coded processes and coded assemblages. These objects, infrastructure, processes and assemblages possess technicity, that is, unfolding or evolutive power to make things happen; the ability to mediate, supplement, augment, monitor, regulate, operate, facilitate, produce collective life. We contend that when the technicity of code is operationalised it transduces one of three forms of hybrid spatial formations: code/space, coded space and backgrounded coded space. These formations are contingent, relational, extensible and scaleless, often stretched out across networks of greater or shorter length. We demonstrate the coded transduction of space through three vignettes – each a day in the life of three people living in London, UK, tracing the technical mediation of their interactions, transactions and mobilities. We then discuss how code becomes the relational solution to five different classes of problems – domestic living, travelling, working, communicating, and consuming

Continuity of Functional Transducers: A Profinite Study of Rational Functions

A word-to-word function is continuous for a class of languages~$\mathcal{V}$ if its inverse maps $\mathcal{V}$_languages to~$\mathcal{V}$. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes. Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. To this end, we develop a robust theory rooted in the standard profinite analysis of regular languages. Since previous algebraic studies of transducers have focused on the sole structure of the underlying input automaton, we also compare the two algebraic approaches. We focus on two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses

Revisiting the Equivalence Problem for Finite Multitape Automata

The decidability of determining equivalence of deterministic multitape automata (or transducers) was a longstanding open problem until it was resolved by Harju and Karhum\"{a}ki in the early 1990s. Their proof of decidability yields a co_NP upper bound, but apparently not much more is known about the complexity of the problem. In this paper we give an alternative proof of decidability, which follows the basic strategy of Harju and Karhumaki but replaces their use of group theory with results on matrix algebras. From our proof we obtain a simple randomised algorithm for deciding language equivalence of deterministic multitape automata and, more generally, multiplicity equivalence of nondeterministic multitape automata. The algorithm involves only matrix exponentiation and runs in polynomial time for each fixed number of tapes. If the two input automata are inequivalent then the algorithm outputs a word on which they differ

Transindividual Equations/Matrices

This writing (thinking-feeling) unfolds (an articulation of) contemporary (dis)embodiment through specified glances at the "(post-)internet,” schizoid relationality, and network/device/identity matrices. Technogenetic (and hormonal) play and transductions for transindividuality on the transparency-obfuscation (or personality/anonymity) binaries arrive as psychotherapeutics for symphonic moving-sensing-feeling-thinking-communicating in a potentiated post-neoliberal matrix (networking). Generated is a set of direction-possibilities for a post-internet-bodied world consumed by a hegemony of individualizations and self-captures, desiring instead towards queerer ceremonies and telepathic forms of (care-)presence/therapy for being-together-alone (individuation). The concurrent/included choreographic and research-artistic work 3M0T1NG{n3tw0rk1ng} peeks into a technogenetic xeno-spacetime containing relating (meta-)selves

Two-way automata and transducers with planar behaviours are aperiodic

We consider a notion of planarity for two-way finite automata and transducers, inspired by Temperley-Lieb monoids of planar diagrams. We show that this restriction captures star-free languages and first-order transductions.Comment: 18 pages, DMTCS submissio