Recent Preprints


  1. Frobenius identities and geometrical aspects of Joyal-Tierney Theorem (with Aleš Pultr)
    July 2024. pdf file
    Abstract: Open and related maps in the point-free context are studied from a consequently geometric perspective: that is, the opens are concrete well-defined subsets, images of localic maps are set-theoretic images f[U], etc.. We present a short proof of Joyal-Tierney Theorem in this setting, a (geometric) characteristic of localic maps that are just complete, and prove that open localic maps also preserve a natural type of sublocales more general than the open ones. A crucial role is played by Frobenius identities that are briefly discussed also in their general aspects.

  2. Continuity and openness of maps on locales by way of Galois adjunctions (with João Areias)
    Preprint 23-27 of DMUC, August 2023. pdf file
    Abstract: This note reviews some adjoint situations, in the algebraic (pointfree) setting of frames and locales, that describe fundamental properties of mappings such as residuation, continuity and openness.

  3. Notes on sublocales and dissolution (with Aleš Pultr)
    Preprint 23-26 of DMUC, August 2023. pdf file
    Abstract: The dissolution (introduced by Isbell in [3], discussed by Johnstone in [5] and later exploited by Plewe in [12, 13]) is here viewed as the relation of the geometry of L with that of the more dispersed T(L)=S(L)op mediated by the natural embedding cL=(a ↦ ↑ a) and its adjoint localic map γL : T(L) → L. The associated image-preimage adjunction (γL)−1[−] ⊣ γL[−] between the frames T(L) and TT(L) is shown to coincide with the adjunction cT(L) ⊣ γT(L) of the second step of the assembly (tower) of L. This helps to explain the role of T(L)=S(L)op as an “almost discrete lift” (sometimes used as a sort of model of the classical discrete lift DL → L as a dispersion going halfway to Booleanness. Consequent use of the concrete sublocales technique simplifies the reasoning. We illustrate it on the celebrated Plewe's Theorem on ultranormality (and ultraparacompactness) of S(L) which becomes (we hope) substantially more transparent.

  4. Notes on the spatial part of a frame (with Igor Arrieta and Aleš Pultr)
    Preprint 23-12 of DMUC, May 2023. pdf file
    Abstract: A locale (frame) L has a largest spatial sublocale generated by the primes (spectrum points), the spatial part SpL. In this paper we discuss some of the properties of the embeddings SpL ⊆ L. First we analyze the behaviour of the spatial parts in the assembly: the points of L and of S(L)op (≅ the congruence frame) are in a natural one-one correspondence while the topologies of SpL and Sp(S(L)op) differ. Then we concentrate on some special types of embeddings of SpL into L, namely in the questions when SpL is complemented, closed, or open. While in the first part L was general, here we need some restrictions (weak separation axioms) to obtain suitable formulas.