## Syntax for Split Preorders (902)

### BibTeX

@MISC{Petrić902syntaxfor,

author = {Zoran Petrić},

title = {Syntax for Split Preorders},

year = {902}

}

### OpenURL

### Abstract

A split preorder is a preordering relation on the disjoint union of two sets, which function as source and target when one composes split preorders. The paper presents by generators and equations the category SplPre, whose arrows are the split preorders on the disjoint union of two finite ordinals. The same is done for the subcategory Gen of SplPre, whose arrows are equivalence relations, and for the category Rel, whose arrows are the binary relations between finite ordinals, and which has an isomorphic image within SplPre by a map that preserves composition, but not identity arrows. It was shown previously that SplPre and Gen have an isomorphic representation in Rel in the style of Brauer. The syntactical presentation of Gen and Rel in this paper exhibits the particular Frobenius algebra structure of Gen and the particular bialgebraic structure of Rel, the latter structure being built upon the former structure in SplPre. This points towards algebraic modelling of various categories motivated by logic, and related categories, for which one can establish coherence with respect Rel and Gen. It also sheds light on the relationship between the notions of Frobenius algebra and bialgebra. The completeness of the syntactical presentations is proved via normal forms, with the normal form for SplPre and Gen being in some sense orthogonal to the composition-free, i.e. cut-free, normal form for Rel. The paper ends by showing that the assumptions for the algebraic structures of SplPre, Gen and Rel cannot be extended with new equations without falling into triviality.

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Citation Context ...ing. The name of Jones monads is derived from the connection of these monads with the monoid Jω of [5] (named with the initial of Jones’ name); this monoid is closely related to monoids introduced in =-=[12]-=- (p. 13), which are called Jones monoids in [15] (as suggested by [5]). It can be shown that the category J of the Jones monad freely generated by a single object is isomorphic to a subcategory of Gen... |

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Citation Context ...), i.e. ¡ ◦! = 1. So the difference with equivalential Frobenius monads is that here symmetry is missing. The name of Jones monads is derived from the connection of these monads with the monoid Jω of =-=[5]-=- (named with the initial of Jones’ name); this monoid is closely related to monoids introduced in [12] (p. 13), which are called Jones monoids in [15] (as suggested by [5]). It can be shown that the c... |

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Citation Context ...owed by the deletion of the part over which the preorders were glued in the pushout (this deletion corresponds to our operation −Bi ). The cospans over the base category of graphs, which one finds in =-=[17]-=-, are more general than our specific cospans, and they do not involve the deletion just mentioned. A split equivalence from A to B is a split preorder from A to B based on a symmetric set of ordered p... |

2 |
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(Show Context)
Citation Context ...t section). The monoids of endomorphisms of Gen, i.e. the monoids of arrows of Gen from n to n, called partition monoids, are involved in the partition algebras 3of V. Jones and P. Martin (see [13], =-=[12]-=- and references therein). We have relied on the categories Rel and Gen in our work on categorial coherence for various fragments of logic, and related structures (see [7], [8], [9], [10], [11], and re... |

1 |
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Citation Context ...he next section). The monoids of endomorphisms of Gen, i.e. the monoids of arrows of Gen from n to n, called partition monoids, are involved in the partition algebras 3of V. Jones and P. Martin (see =-=[13]-=-, [12] and references therein). We have relied on the categories Rel and Gen in our work on categorial coherence for various fragments of logic, and related structures (see [7], [8], [9], [10], [11], ... |