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Grothendieck Inclusion Systems
 APPLIED CATEGORICAL STRUCTURES
"... Inclusion systems have been introduced in algebraic specification theory as a categorical structure supporting the development of a general abstract logicindependent approach to the algebra of specification (or programming) modules. Here we extend the concept of indexed categories and their Grothe ..."
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Inclusion systems have been introduced in algebraic specification theory as a categorical structure supporting the development of a general abstract logicindependent approach to the algebra of specification (or programming) modules. Here we extend the concept of indexed categories and their Grothendieck flattenings to inclusion systems. An important practical significance of the resulting Grothendieck inclusion systems is that they allow the development of module algebras for multilogic heterogeneous specification frameworks. At another level, we show that several inclusion systems in use in some syntactic (signatures, deductive theories) or semantic contexts (models) appear as Grothendieck inclusion systems too. We also study several general properties of Grothendieck inclusion systems.
Foundation of the Rewriting in an Algebra ∗
, 2007
"... This paper includes two main ideas. The first one, rewriting in an algebra, was introduced in [5]. The second one, boolean rewriting, can be found in many papers but we were never able to find a clear comparison with the classic one. We prefer rewriting in an algebra to term rewriting. This is our w ..."
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This paper includes two main ideas. The first one, rewriting in an algebra, was introduced in [5]. The second one, boolean rewriting, can be found in many papers but we were never able to find a clear comparison with the classic one. We prefer rewriting in an algebra to term rewriting. This is our way to give a unique theory of rewriting. If the algebra is free, then we get the term rewriting. If the algebra is a certaine quotient of a free algebra then we get rewriting modulo equations. Rewriting is said to be boolean when the condition of each conditional equation is of boolean sort(in the free algebra it is a boolean term). We prove the classic rewriting is equivalent to boolean rewriting in a specific algebra, therefore, boolean rewriting is more general than the classic one. 1
Weak Inclusion Systems; part 2
"... semantics is given for modularization in [13], based on strong inclusions; no factorization (see Definition 9) is involved, which means that, perhaps, strong inclusions are good enough technical tools to handle complex modularization concepts. Definition 9 hI; Ei is a weak inclusion system of A, or ..."
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semantics is given for modularization in [13], based on strong inclusions; no factorization (see Definition 9) is involved, which means that, perhaps, strong inclusions are good enough technical tools to handle complex modularization concepts. Definition 9 hI; Ei is a weak inclusion system of A, or A has a weak inclusion system hI; Ei, iff I is a subcategory of inclusions of A, E is a subcategory of A having the same objects as A, and every morphism f in A has a unique factorization f = e; i with e 2 E and i 2 I. hI; Ei is called an inclusion system if E contains only epics, and it is called a regular inclusion system if E contains only coequalizers. 2 Example 10 All structures in Example 8 have weak inclusion systems: Set with I the set of inclusions and E the set of surjective functions. It is regular as each surjective function is a retract, so a coequalizer. Top has two interesting weak inclusion systems (see [1]). One is hI 1 ; E 1 i, where I 1 is the set of continuous inclu...
An Axiomatic Approach to Structuring Specifications
"... In this paper we develop an axiomatic approach to structured specifications in which both the underlying logical system and corresponding institution of the structured specifications are treated as abstract institutions, which means two levels of institution independence. This abstract axiomatic app ..."
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In this paper we develop an axiomatic approach to structured specifications in which both the underlying logical system and corresponding institution of the structured specifications are treated as abstract institutions, which means two levels of institution independence. This abstract axiomatic approach provides a uniform framework for the study of structured specifications independently from any actual choice of specification building operators, and moreover it unifies the theory and the model oriented approaches. Within this framework we develop concepts and results about ‘abstract structured specifications ’ such as colimits, model amalgamation, compactness, interpolation, sound and complete proof theory, and pushoutstyle parameterization with sharing, all of them in a top down manner dictated by the upper level of institution independence. 1.
On the Algebra of the Structured Specifications
"... We develop module algebra for structured specifications with model oriented denotations. Our work extends the existing theory with specification building operators for nonprotecting importation modes and with new algebraic rules (most notably for initial semantics) and upgrades the pushoutstyle se ..."
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We develop module algebra for structured specifications with model oriented denotations. Our work extends the existing theory with specification building operators for nonprotecting importation modes and with new algebraic rules (most notably for initial semantics) and upgrades the pushoutstyle semantics of parameterized modules to capture the (possible) sharing between the body of the parameterized modules and the instances of the parameters. We specify a set of sufficient abstract conditions, smoothly satisfied in the actual situations, and prove the isomorphism between the parallel and the serial instantiation of multiple parameters. Our module algebra development is done at the level of abstract institutions, which means that our results are very general and directly applicable to a wide variety of specification and programming formalisms that are rigorously based upon some logical system. 1.
Quasivarieties and Initial Semantics for Hybridized Institutions
"... We define and develop the concept of quasivariety for models of hybrid logics and we apply this for determining initial semantics for classes of hybrid logics theories. The hybrid logic is considered here in a very general sense, internal to abstract institutions (in the sense of the socalled inst ..."
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We define and develop the concept of quasivariety for models of hybrid logics and we apply this for determining initial semantics for classes of hybrid logics theories. The hybrid logic is considered here in a very general sense, internal to abstract institutions (in the sense of the socalled institution theory of Goguen and Burstall). This means our result is applicable to a wide variety of hybrid logics including for example those resulting from the various kinds of combinations between conventional hybrid logics and various other logical systems. 1.