Recent Preprints

1. 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.
   
   
2. 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 c_L=(a ↦ ↑ a) and its adjoint localic map γ_L : T(L) → L. The associated image-preimage adjunction (γ_L)⁻¹[−] ⊣ γ_L[−] between the frames T(L) and TT(L) is shown to coincide with the adjunction c_T(L) ⊣ γ_T(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.
   
   
3. 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.
   
   
4. Some properties of conjunctivity (subfitness) in generalized settings (with M. Andrew Moshier and Aleš Pultr)
   Preprint 22-23 of DMUC, August 2022. pdf file
   Abstract: The property of subfitness used in point-free topology (roughly speaking) to replace the slightly stronger T₁-separation, appeared (as disjunctivity) already in the pioneering Wallman's [15], then practically disappeared to reappear again (conjunctivity, subfitness), until it was in the recent decades recognized as an utmost important condition playing a very special role. Recently, it was also observed that this property (or its dual) appeared independently in general poset setting (e.g. as separativity in connection with forcing). In a recent paper [2], Delzell, Ighedo and Madden discussed it in the context of semilattices. In this article we discuss it on the background of the systems of meet-sets (subsets closed under existing infima) in posets of various generality (smilattices, lattices, distributive lattices, complete lattices) and present parallels of some localic (frame) facts, including a generalized variant of fitness.