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Interconnection of Object Specifications
 Formal Methods and Object Technology
, 1996
"... ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, a ..."
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ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and applying no methods corresponds to the identity of the monoid. A monoid is a set M with an associative binary operation ffl M : M \ThetaM ! M , usually referred to as `multiplication', which has an identity element e M 2 M . If M = (M; ffl M ; e M ) is a monoid, we often write just M for M, and e for e M ; moreover for m;m 0 2 M , we usually write mm 0 instead of m ffl M m 0 . For example, A , the set of lists containing elements of A, together with concatenation ++ : A \ThetaA ! A and the empty list [ ] 2 A , is a monoid. This example is especially important for the material in later sections. A monoid homomorphism is a structure preserving map between the carriers of ...
Interval Valued RImplications and Automorphisms Abstract
"... Interval fuzzy logic is firmly integrated with principles of fuzzy logic theory and interval mathematics. The former provides a complete and inclusive mathematical model of uncertainty from which the foundations of fuzzy control have widened the scope of control theory. The latter models the uncerta ..."
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Interval fuzzy logic is firmly integrated with principles of fuzzy logic theory and interval mathematics. The former provides a complete and inclusive mathematical model of uncertainty from which the foundations of fuzzy control have widened the scope of control theory. The latter models the uncertainty and the errors in numerical computation, leading to selfvalidated methods. Both areas were independently developed in the mid 1960s improving the quantitative analysis of approximations to mathematically exact values, which may not be observable, representable or computable. Interval fuzzy connectives have been described in the terms of the combination of those theories. In this work, the best interval representation is considered for the study of Rimplication in fuzzy logic. Based on the best interval representation, an interval fuzzy Rimplication is obtained as a canonical extension satisfying the optimality property and preserving the same properties satisfied by the fuzzy Rimplication. In addition, commutative diagrams relate fuzzy Rimplications to interval fuzzy Rimplications. This leads to the understanding of how interval automorphisms act on interval Rimplications and generate other interval fuzzy Rimplications. Keywords: Fuzzy logics, Rimplications, interval representations, automorphisms.
APPLICATIONS OF SUPLATTICE ENRICHED CATEGORY
, 1986
"... Grothendieck toposes are studied via the process of taking the associated Slenriched category of category of relations. As a result, pullback and comma toposes are calculated in a new way. The calculations are used to give a new characterization of localic morphisms and to derive interpolation and ..."
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Grothendieck toposes are studied via the process of taking the associated Slenriched category of category of relations. As a result, pullback and comma toposes are calculated in a new way. The calculations are used to give a new characterization of localic morphisms and to derive interpolation and conceptual completeness properties for a certain class of interpretations between geometric theories. A simple characterization of internal suplattices in terms of external Slenriched category theory is given. Contents Page
From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept
"... Abstract. Historical work on the emergence of sheaf theory has mainly concentrated on the topological origins of sheaf cohomology in the period from 1945 to 1950, and on subsequent developments. However, a shift of emphasis both in timescale and disciplinary context can help gain new insight into t ..."
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Abstract. Historical work on the emergence of sheaf theory has mainly concentrated on the topological origins of sheaf cohomology in the period from 1945 to 1950, and on subsequent developments. However, a shift of emphasis both in timescale and disciplinary context can help gain new insight into the emergence of the sheaf concept. This paper concentrates on Henri Cartan’s work in the theory of analytic functions of several complex variables and the strikingly different roles it played at two stages of the emergence of sheaf theory: the definition of a new structure and formulation of a new research programme in19401944; the unexpected integration into sheaf cohomology in 19511952. In order to bring this twostage structural transition into perspective, we will concentrate more specifically on a family of problems, the socalled Cousin problems, from Poincaré (1883) to Cartan. This mediumterm narrative provides insight into two more general issues in the history of contemporary mathematics. First, we will focus on the use of problems in theorymaking. Second, the history ofthe design of structures in geometrically flavoured contexts – such as for the sheaf and fibrebundle structures – which will help provide a more comprehensive view of the structuralist moment, a moment whose algebraic component has so far been the main focus for historical work. Contents
Mass problems and intuitionistic higherorder logic
"... In this paper we study a model of intuitionistic higherorder logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note t ..."
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In this paper we study a model of intuitionistic higherorder logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (∀x∃y A(x, y)) ⇒ ∃w ∀xA(x,wx) and a bounding principle (∀x∃y A(x, y)) ⇒ ∃z ∀x∃y (y ≤T (x, z) ∧ A(x, y)) where x, y, z range over Muchnik reals, w ranges over functions from Muchnik reals to Muchnik reals, and A(x, y) is a formula not containing w or z. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higherorder