On Sabidussi-Fawcett subdirect representation

AbstractSabidussi's representation theorem for symmetric graphs is generalized to fairly general concrete categories. As applications, the lists of the irreducible objects in several cases (for instance, symmetric or directed graphs with or without loops, n-partite graphs, posets) are presented

Axiom TDT_D and the Simmons sublocale theorem

summary:More precisely, we are analyzing some of H. Simmons, S.\,B. Niefield and K.\,I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom $T_D$ for the relation of certain degrees of scatteredness but did not emphasize its role in the relation {between} sublocales and subspaces. S.\,B. Niefield and K.\,I. Rosenthal just mention this axiom in a remark about Simmons' result. In this paper we show that the role of $T_D$ in this question is crucial. Concentration on the properties of $T_D$-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem

On classes of relations and graphs determined by subobjects and factorobjects

AbstractThe classes of relations and graphs determined by subobjects and factorobjects are studied. We investigate whether such classes are closed under products, whether they are finitely generated by products and subobjects and whether a class can be described alternatively by subobjects and factorobjects. This is related to good characterizations

Localic maps constructed from open and closed parts

Assembling a localic map f:L→M from localic maps f_i:S_i→M, i∈J, defined on closed resp. open sublocales (J finite in the closed case) follows the same rules as in the classical case. The corresponding classical facts immediately follow from the behavior of preimages but for obvious reasons such a proof cannot be imitated in the point-free context. Instead, we present simple proofs based on categorical reasoning. There are some related aspects of localic preimages that are of interest, though. They are investigated in the second half of the paper