We develop a theory of minimal models for algebras over a Koszul operad with trivial differential defined over a commutative ring (containing $\mathbb {Q}$ in the symmetric case), not necessarily a field, extending and supplementing the work of Sagave for the associative case. Our minimal models are bigraded and contain a projective resolution of the homology.

We investigate how various forms of bisimulation can be characterised using the technology of logical relations. The approach taken is that each form of bisimulation corresponds to an algebraic structure derived from a transition system, and the general result is that a relation R between two transition systems on state spaces S and T is a bisimulation if and only if the derived algebraic structures are in the logical relation automatically generated from R. We show that this approach works for the original Park–Milner bisimulation and that it extends to weak bisimulation, and branching and semi-branching bisimulation. The paper concludes with a discussion of probabilistic bisimulation, where the situation is slightly more complex, partly owing to the need to encompass bisimulations that are not just relations.

We condense the theory of UTxO blockchains down to a simple and compact set of four type equations (Idealised EUTxO), and to an algebraic characterisation (abstract chunk systems), and exhibit an adjoint pair of functors between them. This gives a novel account of the essential mathematical structures underlying blockchain technology, such as Bitcoin.

In this survey we discuss work of Levin and V’yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V’yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections $\mathcal {A}$ and $\mathcal {B}$ , $\mathcal {A}$ is below $\mathcal {B}$ in this ordering if $\mathcal {A}\setminus \mathcal {B}$ is negligible. The degree structure associated with this ordering, the Levin–V’yugin degrees (or $\mathrm {LV}$ -degrees), can be shown to be a Boolean algebra, and in fact a measure algebra. We demonstrate the interactions of this work with recent results in computability theory and algorithmic randomness: First, we recall the definition of the Levin–V’yugin algebra and identify connections between its properties and classical properties from computability theory. In particular, we apply results on the interactions between notions of randomness and Turing reducibility to establish new facts about specific LV-degrees, such as the LV-degree of the collection of 1-generic sequences, that of the collection of sequences of hyperimmune degree, and those collections corresponding to various notions of effective randomness. Next, we provide a detailed explanation of a complex technique developed by V’yugin that allows the construction of semi-measures into which computability-theoretic properties can be encoded. We provide two examples of the use of this technique by explicating a result of V’yugin’s about the LV-degree of the collection of Martin-Löf random sequences and extending the result to the LV-degree of the collection of sequences of DNC degree.

We generate an algebra on blood phenotypes with multiplication based on the human ABO-blood group inheritance pattern. We assume that gametes are not chosen randomly during meiosis. We investigate some of the properties of this algebra, namely, the set of idempotents, lattice of ideals and the associative enveloping algebra.

This article addresses the question which structures occur as fixed structures of stable structures with a generic automorphism. In particular we give a Galois theoretic characterization. Furthermore, we prove that any pseudofinite field is the fixed field of some model of ACFA, any one-free pseudo-differentially closed field of characteristic zero is the fixed field of some model of DCFA, and that any one-free PAC field of finite degree of imperfection is the fixed field of some model of SCFA.


We study simulation of gate circuits in the infinite algebra of transients recently introduced by Brzozowski and Ésik. A transient is a word consisting of alternating 0s and 1s; it represents a changing signal. In the algebra of transients, gates process transients instead of 0s and 1s. Simulation in this algebra is capable of counting signal changes and detecting hazards. We study two simulation algorithms: a general one that works with any initial state, and a special one that applies only if the initial state is stable. We show that the two algorithms agree in the stable case. We also show that the general algorithm is insensitive to the removal of state variables that are not feedback variables. We prove the sufficiency of simulation: all signal changes occurring in binary analysis are predicted by the general algorithm. Finally, we show that simulation can be more pessimistic than binary analysis, if wire delays are not taken into account. We propose a circuit model that we conjecture to be sufficient for proving the equivalence of simulation and binary analysis for feedback-free circuits.


We prove that an inherently nonfinitely based algebra cannot generate an abelian variety. On the other hand, we show by example that it is possible for an inherently nonfinitely based algebra to generate a strongly solvable variety.