A short proof of representability of fork algebras

In this paper a strong relationship is demonstrated between fork algebras and quasi-projective relation algebras. With the help of Tarski's classical representation theorem for quasi-projective relation algebras, a short proof is given for the representation theorem of fork algebras. As a by-product, we will discuss the difference between relative and absolute representation theorems.

We consider classes of relation algebras expanded with new operations based on the formation of ordered pairs. Examples for such algebras are pairing (or projection) algebras of algebraic logic and fork algebras of computer science. It is proved by Sain and Németi [38] that there is no `strong' representation theorem for all abstract pairing algebras in most set theories including ZFC as well as most non-well-founded set theories. Such a `strong' representation theorem would state that every abstract pairing algebra is isomorphic to a set relation algebra having projection elements which are defined with the help of the real (set theoretic) pairing function. Here we show that, by choosing an appropriate (non-well-founded) set theory as our metatheory, pairing algebras and fork algebras admit such `strong' representation theorems.

The representation theorem for fork algebras was always misunderstood regarding its applications in program construction. Its application was always described as “the portability of properties of the problem domain into the abstract calculus of fork algebras”. In this paper we show that the results provided by the representation theorem are by far more important. Here we show that not only the heuristic power coming from concrete binary relations is captured inside the abstract calculus, but also design strategies for program development can be successfully expressed. This result makes fork algebras a programming calculus by far more powerful than it was previously thought.KeywordsBinary RelationRepresentation TheoremRelation AlgebraProgram ConstructionRelational FrameworkThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U×U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Nemeti, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares.

Topological representation of precontact algebras and a connected version of the Stone Duality Theorem – I

We have seen already how useful a representation theorem (Theorem 11.2) can be applied within the theory (for proving Corollary 11.3, Exercise 11.5 d) and Fubinis Theorem). As in Theorem 11.2 the crucial properties of a functional to be representable as an integral are mono-tonicity and comonotonic additivity (or, as in Greco 1982, a somewhat weaker condition). In Theorem 11.2 the domain of the functional is rather large. In decision situations one often has only restricted information, i.e. the domain of the functional is small. Representation theorems with minimal requirements on the domain are treated here. They are closely related to the extension theorems for set functions of Chapter 2. A further important question (e.g. in decision theory) is under what conditions the representing set function is sub- or supermodular and continuous from below. A corollary of the respective Representation Theorem is the classical Daniell-Stone Representation Theorem, where the representing set function is a measure.

The motivation for this paper comes from the following sources. First, one can observe that the two major concepts underlying the methods of reasoning with incomplete information are the concept of degree of truth of a piece of information and the concept of approximation of a set of information items. We shall refer to the theories employing the concept of degree of truth as to theories of fuzziness and to the theories employing the concept of approximation as to theories of roughness (see [6] for a survey). The algebraic structures relevant to these theories are residuated lattices ([7], [12], [13], [16], [17], [18], [22], [23]) and Boolean algebras with operators ([19], [21], [10], [11]), respectively. Residuated lattices provide an arithmetic of degrees of truth and Boolean algebras equipped with the appropriate operators provide a method of reasoning with approximately determined information. Both classes of algebras have a lattice structure as a basis. Second, both theories of fuzziness and theories of roughness develop generalizations of relation algebras to algebras of fuzzy relations [20] and algebras of rough relations ([4], [5], [9]), respectively. In both classes a lattice structure is a basis. Third, not necessarily distributive lattices with modal operators, which can be viewed as most elementary approximation operators, are recently developed in [24] (distributive lattices with operators are considered in [14] and [25]). With this background, our aim is to begin a systematic study of the classes of algebras that have the structure of a (not necessarily distributive) lattice and, moreover, in each class there are some operators added to the lattice which are relevant for binary relations. Our main interest is in developing relational representation theorems for the classes of lattices with operators under consideration. More precisely, we wish to guarantee that each algebra of our classes is isomorphic to an algebra of binary relations on a set. We prove the theorems of that form by suitably extending the Urquhart representation theorem for lattices ([26]) and the representation theorems presented in [1]. The classes defined in the paper are the parts which put together lead to what might be called lattice-based relation algebras. Our view is that these algebras would be the weakest structures relevant for binary relations. All the other algebras of binary relations considered in the literature would then be their signature and/or axiomatic extensions.Throughout the paper we use the same symbol for denoting an algebra or a relational system and their universes.KeywordsBinary RelationDistributive LatticeRepresentation TheoremResiduated LatticeRelation AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The notions of 2-precontact and 2-contact spaces as well as of extensional (and other kinds) 3-precontact and 3-contact spaces are introduced. Using them, new representation theorems for precontact and contact algebras (satisfying some additional axioms) are proved. They incorporate and strengthen both the discrete and topological representation theorems from [3, 1, 2, 4, 10]. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. In particular, such a bijective correspondence for the RCC systems of [8] is obtained, strengthening in this way the previous representation theorems from [4, 1].

In this paper we present some generalization (at the same time a new and a short proof in the Banach algebra context) of the Weak Spectral Mapping Theorem (WSMT) for non-quasianalytic representations of one-parameter groups.

Let S be a compact space and let X, ∥∥ X be a (real, for simplicity) Banach space. We consider the space C x = C(S,X) of all continuous X-valued functions on S, with the supremum norm ∥∥∞. We prove in this paper a Bochner integral representation theorem for bounded linear operators T: C X → X which satisfy the following condition: x*,y* ∈ X*, f,g ∈ C X : x* of = y* og ⇒ x* o Tf = y* o Tg where X* is the conjugate space of X. In the particular case where X = R, this condition is obviously satisfied by every bounded linear operator T: C R → R and the result reduces to the classical Riesz representation theorem. If the dimension of X is greater than 2, we show by a simple example that not every bounded linear T: C X → X admits an integral representation of the type above, proving that the situation is different from the one dimensional case. Finally we compare our result to another representation theorem where the integration process is performed with respect to an operator valued measure.

In this paper, we study lattice-valued logic and lattice-valued modal logic from an algebraic viewpoint. First, we give an algebraic axiomatization of L-valued logic for a finite distributive lattice L. Then we define the notion of prime L-filters and prove an L-valued version of prime filter theorem for Boolean algebras, by which we show a Stone-style representation theorem for algebras of L-valued logic and the completeness with respect to L-valued semantics. By the representation theorem, we can show that a strong duality holds for algebras of L-valued logic and that the variety generated by L coincides with the quasi-variety generated by L. Second, we give an algebraic axiomatization of L-valued modal logic and establish the completeness with respect to L-valued Kripke semantics. Moreover, it is shown that L-valued modal logic enjoys finite model property and that L-valued intuitionistic logic is embedded into L-valued modal logic of S4-type via Gödel-style translation.

In this paper, we introduce the notion of representation of Bol algebra. We prove an analogue of the classical Engel's theorem and the ex- tension of Ado-Iwasawa theorem for Bol Algebras. We study the category of representations of Bol algebras and show that it is a tensor category. In the case of regular representations of Bol algebras, we give a complete classification of them for all two-dimensional Bol algebras.

At the end of Chapter 4 of the RelMiCS book [11] an application of fork algebras as the basis for a calculus for program construction is outlined. In this paper we make a detailed presentation of the calculus as well as present some examples. We present a methodology for program construction based on the first-order theory of fork algebras. In this theory we will describe program design strategies, for instance case analysis, trivialization, divide-and-conquer and others. Using these strategies, from generic specifications (i.e., parameterized specifications) we will derive parametric algorithms. We will also provide conditions that will help in finding the parameters of the generic algorithms from the parameters in the specifications. We assume the reader is acquainted with the terminology and notation for relation and fork algebras, as well as with their basic properties as they were presented in the RelMiCS book [11].

The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with $${\mathfrak{gl}_N}$$ . We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras $${\mathfrak{sp}_{2n}}$$ . The representations are parameterized by their highest weights satisfying certain dominance-type conditions. In the simplest case of $${\mathfrak{sp}_2}$$ , we give an explicit description of all the representations as tensor products of evaluation modules. We give new proofs of the (well-known) Poincaré–Birkhoff–Witt theorem for the quantum affine algebra and for the twisted q-Yangians in their RTT-presentations. We also reproduce Tarasov’s proof of the classification theorem for finite-dimensional irreducible representations of the quantum affine algebra by relying on its R-matrix presentation.

Luce and Krantz (1971) presented an axiom system for conditional expected utility. In this theory a conditional decision is a function whose domain is a non-null subevent and whose range is a subset of a set of consequences. Given a family of conditional decisions that is closed under unions of decisions whose domains are disjoint and under restrictions to non-null subevents, the second major primitive is an ordering of the family. Axioms were given that are adequate to construct a numerical utility function over decisions and a probability function over events for which the conditional expectation of the utility is order preserving. Several of the axioms are quite complex and seem a bit artificial, and the proof is very long. Here the structure is modified by adding to the set of outcomes a concatenation operation, and the representation theorem is modified by requiring that the utility function be additive over this binary operation as well as exhibiting the expected utility property. The advantages of this pair of changes are, first, it exploits the obvious fact that the union of consequences is itself a consequence; second, it reduces the mathematical burden carried by the set theoretic structure of conditional decisions and, as a result, the axioms can be made much easier to understand; and third, it permits a considerably shorter proof because one can draw more readily on known results. The major drawback of this approach is, of course, that it is inconsistent with the evidence that utility is not additive over consequences - at least, not over increasing amounts of a single good (diminishing marginal utility).