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Complete Axioms for Categorical Fixedpoint Operators
 In Proceedings of 15th Annual Symposium on Logic in Computer Science
, 2000
"... We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the fre ..."
Abstract

Cited by 29 (6 self)
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We give an axiomatic treatment of fixedpoint operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixedpoint operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !continuous functions between !complete pointed partial orders. This possesses a leastfixedpoint oper...
Traced Premonoidal Categories
, 1999
"... Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating trace ..."
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Cited by 7 (0 self)
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Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a wellknown theorem relating traces and Conway operators in cartesian categories.
Partial Conway and iteration semirings
, 712
"... A Conway semiring is a semiring S equipped with a unary operation ∗ : S → S, always called ’star’, satisfying the sum star and product star identities. It is known that these identities imply a Kleene type theorem. Some computationally important semirings, such as N or N rat 〈〈Σ ∗ 〉 〉 of rational po ..."
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A Conway semiring is a semiring S equipped with a unary operation ∗ : S → S, always called ’star’, satisfying the sum star and product star identities. It is known that these identities imply a Kleene type theorem. Some computationally important semirings, such as N or N rat 〈〈Σ ∗ 〉 〉 of rational power series of words on Σ with coefficients in N, cannot have a total star operation satisfying the Conway identities. We introduce here partial Conway semirings, which are semirings S which have a star operation defined only on an ideal of S; when the arguments are appropriate, the operation satisfies the above identities. We develop the general theory of partial Conway semirings and prove a Kleene theorem for this generalization. 1
LeftHanded Completeness
, 2011
"... We give a new, significantly shorter proof of the completeness of the lefthanded star rule of Kleene algebra. The proof reveals the rich interaction of algebra and coalgebra in the theory. 1 ..."
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We give a new, significantly shorter proof of the completeness of the lefthanded star rule of Kleene algebra. The proof reveals the rich interaction of algebra and coalgebra in the theory. 1
Typed Kleene Algebra with Products and Iteration Theories
"... Abstract—We develop a typed equational system that subsumes both iteration theories and typed Kleene algebra in a common framework. Our approach is based on cartesian categories endowed with commutative strong monads to handle nondeterminism. I. ..."
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Abstract—We develop a typed equational system that subsumes both iteration theories and typed Kleene algebra in a common framework. Our approach is based on cartesian categories endowed with commutative strong monads to handle nondeterminism. I.