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Let $X$ be a completely regular Hausdorff space and, as usual, let $C (X)$ denote the ring of real-valued continuous functions on $X$ . The lattice of $z$ -ideals of $C (X)$ has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) $β X$ precisely when $X$ is a $P$ -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a $d$ -ideal if whenever two elements have the same annihilator and...

When the product-preserving functors preserve limits

Věra Trnková (1970)

Commentationes Mathematicae Universitatis Carolinae

Zum Satz von Freyd und Kelly.

Walter Tholen (1978)

Mathematische Annalen

Zur Axiomatik von Kategorien partieller Morphismen

Jürgen Schreckenberger (1987)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

$μ$ -bicomplete categories and parity games

Luigi Santocanale (2002)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

For an arbitrary category, we consider the least class of functors containing the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are definable by $μ$ -terms. We call the category $μ$ -bicomplete if every $μ$ -term defines a functor. We provide concrete examples of such categories and explicitly characterize this class of functors for the category of sets and functions. This goal is achieved...

μ-Bicomplete Categories and Parity Games

Luigi Santocanale (2010)

RAIRO - Theoretical Informatics and Applications

For an arbitrary category, we consider the least class of functors containing the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are definable by μ-terms. We call the category μ-bicomplete if every μ-term defines a functor. We provide concrete examples of such categories and explicitly characterize this class of functors for the category of sets and functions. This goal is achieved...