# Complements in Distributive Allegories

@inproceedings{Winter2009ComplementsID, title={Complements in Distributive Allegories}, author={Michael Winter}, booktitle={RelMiCS}, year={2009} }

It is known in topos theory that the axiom of choice implies that the topos is Boolean. In this paper we want to prove and generalize this result in the context of allegories. In particular, we will show that partial identities do have complements in distributive allegories with relational sums and total splittings assuming the axiom of choice. Furthermore, we will discuss possible modifications of the assumptions used in that theorem.

## 5 Citations

### Cardinal Addition in Distributive Allegories

- MathematicsRelMiCS
- 2009

This paper is going to define addition on cardinalities and investigate its basic properties, and extend the abstract approach to the size of a relation based on a cardinality function.

### Point Axioms in Dedekind Categories

- Economics, PhilosophyRAMiCS
- 2012

This paper summarises interrelations of axioms of Dedekind categories to connect functional ideas to set-theoretical intuition.

### Point axioms and related conditions in Dedekind categories

- Economics, PhilosophyJ. Log. Algebraic Methods Program.
- 2015

### Investigating and Computing Bipartitions with Algebraic Means

- Mathematics, Computer ScienceRAMiCS
- 2015

Using Dedekind categories as an algebraic structure for (binary) set-theoretic relations without complements, we present purely algebraic definitions of “to be bipartite” and “to possess no odd…

### Using relation-algebraic means and tool support for investigating and computing bipartitions

- MathematicsJ. Log. Algebraic Methods Program.
- 2017

## References

SHOWING 1-10 OF 17 REFERENCES

### Cardinal Addition in Distributive Allegories

- MathematicsRelMiCS
- 2009

This paper is going to define addition on cardinalities and investigate its basic properties, and extend the abstract approach to the size of a relation based on a cardinality function.

### Axiom of choice and complementation

- Mathematics
- 1975

It is shown that an intuitionistic model of set theory with the axiom of choice has to be a classical oneO A topos 6 is a category which has finite limits (i.e. finite products, intersections and a…

### Goguen Categories

- Mathematics
- 2004

This paper is a survey of the theory of Goguen categories which establishes a suitable categorical description of L-fuzzy relations, i.e., of relations taking values from an arbitrary complete…

### REVIEWS-Sketches of an elephant: A topos theory compendium

- Mathematics
- 2003

A1 Regular and Cartesian Closed Categories A2 Toposes - Basic Theory A3 Allegories A4 Geometric Morphisms - Basic Theory B1 Fibrations and Indexed Categories B2 Internal and Locally Internal…

### The derivation of identities involving projection functions

- Mathematics
- 1993

The \unsharpness problem" is solved by the construction of a nite Although this equation fails in general, it does hold under slightly stronger hypotheses. x1 In recent years various…

### Goguen Categories: A Categorical Approach to L-fuzzy Relations

- Computer Science
- 2007

This book introduces Goguen categories and provides a comprehensive study of these structures including their representation theory, and the definability of norm-based operations.

### A Formalization Of Set Theory Without Variables

- Mathematics
- 1987

The formalism $\mathcal L$of predicate logic The formalism $\mathcal L^+$, a definitional extension of $\mathcal L$ The formalism $\mathcal L^+$ without variables and the problem of its equipollence…

### Relations and Graphs: Discrete Mathematics for Computer Scientists

- Computer Science
- 1993

This book is the first to explain how to use relational and graph-theoretic methods systematically in computer science, and develops a formal framework of relational algebra with respect to applications to a diverse range of problem areas.

### Relational Methods in Computer Science

- Computer ScienceAdvances in Computing Sciences
- 1997

This book discusses Relational Formalisation of Nonclassical Logics, Logic, Language, and Information, and its applications in Programs and Datatypes, and other Application Areas.