Abstract Given a locale L , the ordered collection $$\textsf{S}_c(L)$$ S c ( L ) of joins of closed sublocales forms a frame—somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of L , where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc. A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether $$\textsf{S}_c(L)$$ S c ( L ) is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale L such that $$\textsf{S}_c(L)$$ S c ( L ) is not a coframe. The main challenge in such question lies in the difficulty of understanding exact infima in $$\textsf{S}_c(L)$$ S c ( L ) ; we circumvent this by analysing a certain separation property satisfied by $$\textsf{S}_c(L)$$ S c ( L ) .

Abstract NFTs or non-fungible tokens are digital assets stored on a blockchain. They can be traded or exchanged for money, cryptocurrencies or other NFTs. Examples include works of art and digital or other tokenised collectables. An important determinant of price for collectables is rarity within a collection. Many trading platforms offer to rank items in terms of rarity but rankings differ considerably and, often, little explanation is given of the methods used. This paper provides a mathematical framework for the analysis of a comprehensive class of collections. It examines individual and joint distributions of attributes over such collections, and shows how these can be combined to provide a rarity ranking for all items in the collection. There is, however, only a limited range of methods that give consistent results over different collections. These are identified as belonging to a one-parameter family of ranking functions. Each gives to every item of a collection a rarity score that is directly comparable between collections. Despite taking account of all possible combinations of attributes when ranking, the method is nonetheless computationally feasible.

Abstract We prove that the incomparability orthoset of a finite poset is Dacey if and only if the poset is N-free. We give a characterization of finite posets with compatible incomparability orthosets.