Absolutely Free Multialgebras

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in exploring the foundations of multialgebras applied to the study of logic systems. It is well known from universal algebra that, for every signature \(\Sigma\), there exist algebras over \(\Sigma\) which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of \(\Sigma\)-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of \(\Sigma\)-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor \(\mathcal{U}\), from the category of \(\Sigma\)-multialgebras to Set, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that \(\mathcal{U}\) does not have a left adjoint.

A semiring-based trace semantics for processes with applications to information leakage analysis

We propose a framework for reasoning about program security building on language-theoretic and coalgebraic concepts. The behaviour of a system is viewed as a mapping from traces of high (unobservable) events to low (observable) events: the less the degree of dependency of low events on high traces, the more secure the system. We take the abstract view that low events are drawn from a generic semiring, where they can be combined using product and sum operations; throughout the paper, we provide instances of this framework, obtained by concrete instantiations of the underlying semiring. We specify systems via a simple process calculus, whose semantics is given as the unique homomorphism from the calculus into the set of behaviours, i.e. formal power series, seen as a final coalgebra. We provide a compositional semantics for the calculus in terms of rational operators on formal power series and show that the final and the compositional semantics coincide. This compositional, syntax-driven framework lays a foundation for automation and abstraction of a quantified approach to flow security of system specifications.

A factorization of dual prehomomorphisms and expansions of inverse semigroups

For any inverse semigroup S we construct an inverse semigroup S(S), which has the following universal property with respect to dual prehomomorphisms from S: there is an injective dual prehomomorphism ιS: S → S(S) such that for each dual prehomomorphism θ from S into an inverse semigroup T there exists a unique homomorphism θ* : S(S) → T with ιSθ* = θ. If we restrict the class of dual prehomomorphisms under consideration to order preserving ones, S(S) may be replaced by a certain homomorphic image Ŝ(S) which can be viewed as a natural generalization of the Birget-Rhodes prefix expansion for groups [4] to inverse semigroups. Recently, Lawson, Margolis and Steinberg [8] have given an alternative description of Ŝ(S) which is based on O’Carroll’s theory of idempotent pure congruences [11]. It should be noted that our ideas can be used to simplify some of their arguments.

A Class of Homomorphism Theories for Groupoids

In a well-behaved homomorphism theory for a class of algebraic systems certain “closed objects” relative to a given G ∈ are distinguished which act as kernels of homomorphisms. For example, if is the class of groups then the closed objects relative to a given group G are the normal subgroups of G; if is the class of semigroups with zero element then one can devise a homomorphism theory in which the closed objects relative to a given S ∈ are the ideals of S[cf. Rees (3)]; in the class of groupoids one may define the closed objects relative to a given groupoid G to be the congruence relations on G, that is, subsets π⊆G×G which are equivalence relations having the property that (x1y1, x2y2) ∈ π whenever (x1, x2), (y1, y2) ∈ π. Given such a closed object N relative to G there exists a “factor” system G/N and a (canonical) homomorphism η: G→G/N characterised by the property: If σ G→H is a homomorphism with kernel N then there is a unique homomorphism : G/N→H such that . η = σ and the kernel of is trivial in the sense that the kernel of is the unique smallest closed object relative to G/N.