Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. This method is applied to devising a new paradigm for general topology, called Abstract Stone Duality. We express the duality between algebra and...

We give substance to the motto “every partial algebra is the colimit of its total subalgebras” by proving it for partial Boolean algebras (including orthomodular lattices), the new notion of partial C*-algebras (including noncommutative C*-algebras), and variations such as partial complete Boolean algebras and partial AW*-algebras. Both pairs of results are related by taking projections. As cor...

We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that are compact in one and closed in the other topology. This is in analogy to the celebrated Hofmann-M...

In 1937 Marshall Stone extended his celebrated representation theorem for Boolean algebras to distributive lattices. In modern terminology, the representing topological spaces are zero-dimensional stably compact, but typically not Hausdorff. In 1970, Hilary Priestley realised that Stone’s topology could be enriched to yield orderdisconnected compact ordered spaces. In the present paper, we gene...

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (innuenced by and abstracting over both Goldblatt's 17] and Urquhart's 46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial a...

We unify several extensions of the classic Stone duality due to Grätzer, Hofmann–Lawson and Jung–Sünderhauf. Specifically, we show that $$\cup $$ -bases locally compact sober spaces are dual $$\prec -distributive $$\vee -predomains, where is a relation representing containment.

In this paper we provide a Stone style duality for monotone semilattices by using the topological developed in S. Celani, L.J. González (Appl Categ Struct 28:853–875, 2020) together with description of their canonical extension. As an application obtain characterization congruences means lower-Vietoris-type topologies.

By a strong conceptual completeness (XC) statement for a logical doctrine we mean an assertion that says that any theory of the doctrine can be recovered from an appropriate structure formed by the models of the theory. The expression ‘logical doctrine’ is used here as in [9] to mean a specific selection of logical operations giving rise to a notion of theory, in the form of a structured catego...