Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 20 1 2024 01 01 Notes on the spatial part of a frame 105 129 104138 10.48308/cgasa.2023.233584.1435 EN Igor Arrieta School of Computer Science, University of Birmingham, B15 2TT Birmingham, UK 0000-0002-5319-4916 Jorge Picado Department of Mathematics University of Coimbra PORTUGAL 0000-0001-7837-1221 Ales Pultr Department of Applied Mathematics and ITI, MFF, Charles University, Malostranské ném. 24, 11800 Praha 1, Czech Republic 0000-0002-9308-3700 Journal Article 2023 10 27 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 (∼=<br />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 https://cgasa.sbu.ac.ir/article_104138_c6a276463be68e8c90c59f4dc7645cf1.pdf