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Reductions between cardinal characteristics of the continuum
 In Set Theory (Boise, ID, 1992–1994), 31–49. Contemp. Math., 192, AMS
, 1996
"... Abstract. We discuss two general aspects of the theory of cardinal characteristics of the continuum, especially of proofs of inequalities between such characteristics. The first aspect is to express the essential content of these proofs in a way that makes sense even in models where the inequalities ..."
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Abstract. We discuss two general aspects of the theory of cardinal characteristics of the continuum, especially of proofs of inequalities between such characteristics. The first aspect is to express the essential content of these proofs in a way that makes sense even in models where the inequalities hold trivially (e.g., because the continuum hypothesis holds). For this purpose, we use a Borel version of Vojtáˇs’s theory of generalized GaloisTukey connections. The second aspect is to analyze a sequential structure often found in proofs of inequalities relating one characteristic to the minimum (or maximum) of two others. Vojtáˇs’s maxmin diagram, abstracted from such situations, can be described in terms of a new, highertype object in the category of generalized GaloisTukey connections. It turns out to occur also in other proofs of inequalities where no minimum (or maximum) is mentioned. 1.
On the Compositionality and Analysis of Algebraic HighLevel Nets
 Research Report A16, Digital Systems Laboratory
, 1991
"... : This work discusses three aspects of net theory: compositionality of nets, analysis of nets and highlevel nets. Net theory has often been criticised for the difficulty of giving a compositional semantics to a net. In this work we discuss this problem form a category theoretic point of view. In ca ..."
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: This work discusses three aspects of net theory: compositionality of nets, analysis of nets and highlevel nets. Net theory has often been criticised for the difficulty of giving a compositional semantics to a net. In this work we discuss this problem form a category theoretic point of view. In category theory compositionality is represented by colimits. We show how a highlevel net can be mapped into a lowlevel net that represents its behaviour. This construction is functorial and preserves colimits, giving a compositional semantics for these highlevel nets. Using this semantics we propose some proof rules for compositional reasoning with highlevel nets. Linear logic is a recently discovered logic that has many interesting properties. From a net theoretic point of view its interest lies in the fact that it is able to express resources in an analogous manner to nets. Several interpretations of Petri nets in terms of linear logic exist. In this work we study a model theoretic inter...