Infinite combinatorial issues raised by lifting problems in universal algebra

The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critical point is large), large free set theorems and the classical notation (k,r,l){\to}m (for proving that a critical point is small). In the middle, we find l-lifters of posets and the relation (k, < l){\to}P, for infinite cardinals k and l and a poset P.Comment: 22 pages. Order, to appea

Large semilattices of breadth three

A 1984 problem of S.Z. Ditor asks whether there exists a lattice of cardinality aleph two, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin's Axiom restricted to collections of aleph one dense subsets in posets of precaliber aleph one, (2) the existence of a gap-1 morass. In particular, the existence of such a lattice is consistent with ZFC, while the non-existence of such a lattice implies that omega two is inaccessible in the constructible universe. We also prove that for each regular uncountable cardinal $\kappa$ and each positive integer n, there exists a join-semilattice L with zero, of cardinality $\kappa^{+n}$ and breadth n+1, in which every principal ideal has less than $\kappa$ elements.Comment: Fund. Math., to appea

Representation of algebraic distributive lattices with N1 compact elements as ideal lattices of regular rings

We prove the following result: Theorem. Every algebraic distributive lattice D with at most N1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the N1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively, that L has a largest element. This extends several known results of G. M. Bergman, A. P. Huhn, J. Tuma, and of a joint work of G. Grätzer, H. Lakser, and the author, and it solves Problem 2 of the survey paper [10]. The main tool used in the proof of our result is an amalgamation theorem for semilattices and algebras (over a given division ring), a variant of previously known amalgamation theorems for semilattices and lattices, due to J. Tuma, and G. Grätzer, H. Lakser, and the author

Agoric computation: trust and cyber-physical systems

In the past two decades advances in miniaturisation and economies of scale have led to the emergence of billions of connected components that have provided both a spur and a blueprint for the development of smart products acting in specialised environments which are uniquely identifiable, localisable, and capable of autonomy. Adopting the computational perspective of multi-agent systems (MAS) as a technological abstraction married with the engineering perspective of cyber-physical systems (CPS) has provided fertile ground for designing, developing and deploying software applications in smart automated context such as manufacturing, power grids, avionics, healthcare and logistics, capable of being decentralised, intelligent, reconfigurable, modular, flexible, robust, adaptive and responsive. Current agent technologies are, however, ill suited for information-based environments, making it difficult to formalise and implement multiagent systems based on inherently dynamical functional concepts such as trust and reliability, which present special challenges when scaling from small to large systems of agents. To overcome such challenges, it is useful to adopt a unified approach which we term agoric computation, integrating logical, mathematical and programming concepts towards the development of agent-based solutions based on recursive, compositional principles, where smaller systems feed via directed information flows into larger hierarchical systems that define their global environment. Considering information as an integral part of the environment naturally defines a web of operations where components of a systems are wired in some way and each set of inputs and outputs are allowed to carry some value. These operations are stateless abstractions and procedures that act on some stateful cells that cumulate partial information, and it is possible to compose such abstractions into higher-level ones, using a publish-and-subscribe interaction model that keeps track of update messages between abstractions and values in the data. In this thesis we review the logical and mathematical basis of such abstractions and take steps towards the software implementation of agoric modelling as a framework for simulation and verification of the reliability of increasingly complex systems, and report on experimental results related to a few select applications, such as stigmergic interaction in mobile robotics, integrating raw data into agent perceptions, trust and trustworthiness in orchestrated open systems, computing the epistemic cost of trust when reasoning in networks of agents seeded with contradictory information, and trust models for distributed ledgers in the Internet of Things (IoT); and provide a roadmap for future developments of our research