Notes on the spatial part of a frame

Document Type : Research Paper


1 School of Computer Science, University of Birmingham, B15 2TT Birmingham, UK

2 Department of Mathematics University of Coimbra PORTUGAL

3 Department of Applied Mathematics and ITI, MFF, Charles University, Malostranské ném. 24, 11800 Praha 1, Czech Republic



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


Main Subjects

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