Partial Monoids Research Articles

Law and Order for Typestate with Borrowing

Typestate systems are notoriously complex as they require sophisticated machinery for tracking aliasing. We propose a new, transition-oriented foundation for typestate in the setting of impure functional programming. Our approach relies on ordered types for simple alias tracking and its formalization draws on work on bunched implications. Yet, we support a flexible notion of borrowing in the presence of typestate. Our core calculus comes with a notion of resource types indexed by an ordered partial monoid that models abstract state transitions. We prove syntactic type soundness with respect to a resource-instrumented semantics. We give an algorithmic version of our type system and prove its soundness. Algorithmic typing facilitates a simple surface language that does not expose tedious details of ordered types. We implemented a typechecker for the surface language along with an interpreter for the core language.

Classification of variants of partial Brauer monoids

A variant of a semigroup S with respect to an element a ? S is the semigroup S a = (S, *a), where x *a y = xay for any x, y ? S. Here, a is the sandwich element of Sa. In this article, we study variants of the partial Brauer monoid PBn for n ? N. We give the classification of these variants in the case when the rank of the sandwich element is nonzero.

Post-Quantum Cryptography: generalized ElGamal cipher over GF(251^8)

Post-Quantum Cryptography (PQC) attempts to find cryptographic protocols resistant to attacks by means of for instance Shor’s polynomial time algorithm for numerical field problems like integer factorization (IFP) or the discrete logarithm (DLP). Other aspects are the backdoors discovered in deterministic random generators or recent advances in solving some instances of DLP. The use of alternative algebraic structures like non-commutative or non-associative partial groupoids, magmas, monoids, semigroups, quasigroups or groups, are valid choices for these new kinds of protocols. In this paper, we focus in an asymmetric cipher based on a generalized ElGamal non-arbitrated protocol using a non-commutative general linear group. The developed protocol forces a hard subgroup membership search problem into a non-commutative structure. The protocol involves at first a generalized Diffie-Hellman key interchange and further on the private and public parameters are recursively updated each time a new cipher session is launched. Security is based on a hard variation of the Generalized Symmetric Decomposition Problem (GSDP). Working with $GF(251^8)$ a 64-bits security is achieved, and if $GF(251^{16})$ is chosen, the security rises to 127-bits. An appealing feature is that there is no need for big number libraries as all arithmetic if performed in $\Z_{251}$ and therefore the new protocol is particularly useful for computational platforms with very limited capabilities like smartphones or smartcards.

2-Segal sets and the Waldhausen construction

It is known by results of Dyckerhoff–Kapranov and of Gálvez-Carrillo–Kock–Tonks that the output of the Waldhausen S•-construction has a unital 2-Segal structure. Here, we prove that a certain S•-functor defines an equivalence between the category of augmented stable double categories and the category of unital 2-Segal sets. The inverse equivalence is described explicitly by a path construction. We illustrate the equivalence for the known examples of partial monoids, cobordism categories with genus constraints and graph coalgebras.

On Monoids in the Category of Sets and Relations

The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every homset $\mathbf{Rel}(A,B)$ is a poset with respect to inclusion. We examine the 2-category of monoids $\mathbf{RelMon}$ in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.

Partial monoid actions and a class of restriction semigroups

We study classes of proper restriction semigroups determined by properties of partial actions underlying them. These properties include strongness, antistrongness, being defined by a homomorphism, being an action etc. Of particular interest is the class determined by homomorphisms, primarily because we observe that its elements, while being close to semidirect products, serve as mediators between general restriction semigroups and semidirect products or W-products in an embedding-covering construction. It is remarkable that this class does not have an adequate analogue if specialized to inverse semigroups. F-restriction monoids of this class, called ultra F-restriction monoids, are determined by homomorphisms from a monoid T to the Munn monoid of a semilattice Y. We show that these are precisely the monoids Y⁎mT considered by Fountain, Gomes and Gould. We obtain a McAlister-type presentation for the class given by strong dual prehomomorphisms and apply it to construct an embedding of ultra F-restriction monoids, for which the base monoid T is free, into W-products of semilattices by monoids. Our approach yields new and simpler proofs of two recent embedding-covering results by Szendrei.

The Homology of Partial Monoid Actions and Petri Nets

The aim of this paper is to study the homology theory of partial monoid actions and apply it to computing the homology groups of mathematical models for concurrency. We study the Baues–Wirsching homology groups of a small category associated with a partial monoid action on a set. We prove that these groups can be reduced to the Leech homology groups of the monoid. For a trace monoid with a partial action on a set, we build a complex of free Abelian groups for computing the homology groups of this small category. It allows us to solve the problem posed by the author on the construction of an algorithm to computing the homology groups of elementary Petri nets. We describe the algorithm and give examples of computing the homology groups.

Abelian extensions of partially ordered partial monoids

In this paper we introduce the notion of a partially ordered partial monoid (po PM), and we study extensions of cancellative po PMs by abelian groups. We concentrate on the so-called central extensions, and prove that every such extension is an F-product of a po PM by an abelian group defined by a cocycle $$f.$$ Like in partially ordered groups, the extension can be ordered by means of special sets. We also compare extensions of po PMs with the extensions of their universal groups.

Quotients of dimension effect algebras

We show that the quotient of a dimension effect algebra by its dimension equivalence relation is a unital bounded lattice-ordered positive partial abelian monoid that satisfies a version of the Riesz decomposition property. For a dimension effect algebra of finite type, the quotient is a centrally orthocomplete Stone–Heyting MV-effect algebra; moreover, an orthocomplete effect algebra in which equality is a dimension equivalence relation is the same thing as a complete Stone–Heyting MV-effect algebra.

Special ideals in partial abelian monoids

Under some conditions, the special congruences of partial abelian monoid are those induced by the special ideals, and a class of special ideals of partial abelian monoid has some upper and lower bound properties.

An Algebraic Framework for Defining Behaviours of Concurrent Systems. Part 1: The Constructive Presentation

The paper is the first part of a two-part paper that contributes with a concept of a process viewed as a model of a run of a phenomenon (discrete, continuous, or of a mixed type), with operations allowing to define complex processes in terms of their components, with the respective algebras, and with the idea of using the formal tools thus obtained to describe the behaviours of concurrent systems. A process may have an initial state (a source), a final state (a target), or both. A process can be represented by a partially ordered multiset. Processes of which one can be a continuation of the other can be composed sequentially. Independent processes, i.e. processes which do not disturb each other, can be composed in parallel. Processes may be prefixes, i.e. independent components of initial segments of other processes. Processes in a universe of objects and operations on such processes form a partial algebra, called algebra of processes. Algebras of processes are partial categories with respect to the sequential composition, and partial monoids with respect to the parallel composition. Algebras of processes can be used to define behaviours of concurrent systems. The behaviour of a system can be defined as the set of possible processes of this system with a structure on this set. Such a set is prefix-closed. The structure on this set reflects the prefix order and, possibly, specific features of the behaviour like observability, the relation to flow of real time, etc. Algebras of processes can also be used to define behaviours, to define operations on behaviours similar to those in the existing calculi of behaviours, and to define random behaviours. The first part of the whole paper investigates algebras of processes and their applications to describing behaviours of systems. In the second part the properties of algebras of processes described in the first part are regarded as axioms defining a class of abstract partial algebras, called behaviouroriented algebras, and they initiate a theory of such algebras.

An Algebraic Framework for Defining Behaviours of Concurrent Systems. Part 2: The Axiomatic Presentation

The paper is the second part of two-part paper that contributeswith a concept of a process, with operations allowing to define complex processes in terms of their components, with the respective algebras, and with the idea of using the formal tools thus obtained to describe the behaviours of concurrent systems. In the first part a universal model of a process have been introduced and operations on processes have been defined. A process is viewed as a model of a run of a system (discrete, continuous, or of a mixed type). A process may have an initial state (a source), a final state (a target), or both. A process can be represented by a partially ordered multiset. Processes of which one is a continuation of the other can be composed sequentially. Independent processes, can be composed in parallel. Processes may be prefixes, i.e. independent components of initial segments of other processes. It has been shown that processes in a universe of objects and operations on such processes form a partial algebra, called algebra of processes, that is a specific partial category with respect to the sequential composition, and a specific partial monoid with respect to the parallel composition. In the second part the properties of algebras of processes described in the first part are regarded as axioms defining a class of abstract partial algebras, called behaviour-oriented algebras, and properties of such algebras are investigated. In particular, it is shown how some of the behaviour-oriented algebras can be represented as algebras of processes, and how to use them to describe phenomena with states and processes provided with specific structures.

Gödel incompleteness in AF C*-algebras

For any (possibly, non-unital) AF C*-algebra A with comparability of projections, let D(A) be the Elliott partial monoid of A, and G(A) the dimension group of A with scale D(A). For D ⊆ D(A) a generating set of G(A) let 𝒫 be the set of all formal inequalities a1 + ⋯ + ak ≤ b1 + ⋯ + bl satisfied by G(A), for any ai, bj ∈ D. By Elliott's classification, 𝒫 together with the list of all sums a1 + ⋯ + ak ∈ D(A) uniquely determines A. Can 𝒫 be Gödel incomplete, i.e., effectively enumerable but undecidable? We give a negative answer in case D is finite, and a positive answer in the infinite case. We also show that the range of the map A ↦ D(A) precisely consists of all countable partial abelian monoids satisfying the following three conditions: (i) a + b = a + c ⇒ b = c, (ii) a + b = 0 ⇒ a = b = 0 and (iii) ∀a, b ∈ E ∃c ∈ E such that either a + c = b or b + c = a.

Models and Separation Logics for Resource Trees

In this article, we propose a new data structure, called resource tree, that is a node-labelled tree in which nodes contain resources which belong to a partial monoid. We define the resource tree model and a new separation logic (BI-Loc) that extends the Bunched Implications logic (BI) with a modality for locations. In addition, we consider quantifications on locations and paths and then we study decidability by model-checking in these models and logics. Moreover, we define a language to deal with resource trees and also an assertion logic derived from BI-Loc. Then soundness and completeness issues are studied, and we show how the model and its associated language can be used to manage heap structures and also permission accounting.

The support of a uniform semigroup valued measure

In this paper we introduce the notion of orthogonality for arbitrary families of elements of a partial abelian monoid, we study uniform semigroup valued measures on a partial abelian monoid, and we establish interesting results about the existence of the support of such measures.

Labeled configuration spaces and group completions

Given a pair of a partial abelian monoid M and a pointed space X, let CM(ℝ∞, X) denote the configuration space of finite distinct points in ℝ∞ parametrized by the partial monoid X ∧ M. In this note we will show that if M is embedded in a topological abelian group and if we put ±M = {a − b | a, b ∈ M} then the natural map CM(ℝ∞, X) → C±M(ℝ∞, X) induced by the inclusion M ⊂ ±M is a group completion. This result can be applied to show that for any finite set M such that {0} ⊊ M ⊂ ℤ, CM(ℝ∞, X) is weakly equivalent to the infinite loop space Ω∞Σ∞X if X is connected.

Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes

For any Coxeter system (W, S), the group W acts naturally on the complement of the associated complex hyperplane arrangement. By the well-known conjecture, the orbit space of this action is the classifying space of the corresponding Artin group. We describe some properties of configuration spaces of particles labeled by elements of a partial monoid and use them to prove that the orbit space mentioned in the conjecture is the classifying space of the positive Artin monoid. In particular, the conjecture reduces to a problem concerning the group completion of this monoid.

Universal contingent claims in a general market environment and multiplicative measures: Examples and applications

We present and further develop the concept of a universal contingent claim introduced by the author in 1995. This concept provides a unified framework for the analysis of a wide class of financial derivatives. A universal contingent claim describes the time evolution of a contingent payoff. In the simplest case of a European contingent claim, this time evolution is given by a family of nonnegative linear operators, the valuation operators. For more complex contingent claims, the time evolution that is given by the valuation operators can be interrupted by discrete or continuous activation of external influences that are described by, generally speaking, nonlinear operators, the activation operators. For example, Bermudan and American contingent claims represent discretely and continuously activated universal contingent claims with the activation operators being the nonlinear maximum operators. We show that the value of a universal contingent claim is given by a multiplicative measure introduced by the author in 1995. Roughly speaking, a multiplicative measure is an operator-valued (in general, an abstract measure with values in a partial monoid) function on a semiring of sets which is multiplicative on the union of disjoint sets. We also show that the value of a universal contingent claim is determined by a, generally speaking, impulsive semilinear evolution equation.

Extensions of MV-algebras

We introduce MV-extensions of MV-algebras and their ideals as a special kind of extensions in the category of partial abelian monoids and show their relations with the extensions of the corresponding l-groups.

Quotients of partial abelian monoids and the Riesz decomposition property

Partial abelian monoids (PAMs) are structures ( $ P; \bot, \oplus, 0 $ ), where $ \oplus $ is a partially defined binary operation with domain $ \bot $ , which is commutative and associative in a restricted sense, and 0 is the neutral element. PAMs with the Riesz decomposition properties and binary relations with special properties on PAMs are studied. Relations with abelian groups, dimension equivalence and K 0 for AF C*-algebras are discussed.