A Brauerian representation of split preorders

Abstract

Split preorders are preordering relations on a domain whose composition is defined in a particular way by splitting the domain into two disjoint subsets. These relations and the associated composition arise in categorial proof theory in connection with coherence theorems. Here split preorders are represented isomorphically in the category whose arrows are binary relations, where composition is defined in the usual way. This representation is related to a classical result of representation theory due to Richard Brauer.Mathematics Subject Classification (2000): 03F07, 18A15, 16G99, 05C20

Coherence for star-autonomous categories

“…that * is associative and that 1 n is an identity arrow) is proved in [10] and [11]. This proof is obtained via an isomorphic representation of Br in the category Rel, whose objects are the finite ordinals and whose arrows are all the relations between these objects.…”

Coherence for modalities

“…We do not draw in such pictures the loops corresponding to the pairs (x, x). Composition of arrows is defined, roughly speaking, as the transitive closure of the union of the two relations composed, where we omit the ordered pairs one of whose members is in the middle (see [10], Section 2, and [11], Section 2, for a detailed definition). For example, the split equivalences R 1 and R 2 corresponding to the following two pictures…”

“…In the first part of the paper, for categories in Sections 2-5, these relations are either relations between finite ordinals or split equivalences between finite ordinals. A split equivalence is an equivalence relation on the union of two disjoint source and target sets (see [14], Section 2.3, [10] and [11]), which here we take to be finite ordinals. For the categories in Sections 6-8, our relations are always split equivalences between finite ordinals.…”

Simplicial Endomorphisms

“…What we do should be clear from the following example. We draw as follows the frieze {[2, 3], [4,5], [10,11]…”

“…to ascertain that friezes make a monoid, we need a more formal definition of composition. Formally, we may define composition of friezes either in a geometrical style (see [8]), or in a set-theoretical style, as a peculiar composition of equivalence relations (see [9] or [10]). We don't have space here to go into these formal matters, which have already been treated elsewhere.…”