### Abstract

Original language | English |
---|---|

Pages (from-to) | 367-396 |

Journal | Order : A Journal on the Theory of Ordered Sets and its Applications |

Volume | 9 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1992 |

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### Keywords

- Ordering
- axiomatization
- closure properties
- transitivity

### Cite this

*Order : A Journal on the Theory of Ordered Sets and its Applications*,

*9*(4), 367-396. https://doi.org/10.1007/BF00420357

}

*Order : A Journal on the Theory of Ordered Sets and its Applications*, vol. 9, no. 4, pp. 367-396. https://doi.org/10.1007/BF00420357

**Towards an axiomatization of orderings.** / Storcken, T.; de Swart, Harrie.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Towards an axiomatization of orderings

AU - Storcken, T.

AU - de Swart, Harrie

PY - 1992

Y1 - 1992

N2 - A set of six axioms for sets of relations is introduced. All well-known sets of specific orderings, such as linear and weak orderings, satisfy these axioms. These axioms impose criteria of closedness with respect to several operations, such as concatenation, substitution and restriction. For operational reasons and in order to link our results with the literature, it is shown that specific generalizations of the transitivity condition give rise to sets of relations which satisfy these axioms. Next we study minimal extensions of a given set of relations which satisfy the axioms. By this study we come to the fundamentals of orderings: They appear to be special arrangements of several types of disorder. Finally we notice that in this framework many new sets of relations have to be regarded as a set of orderings and that it is not evident how to minimize the number of these new sets of orderings.

AB - A set of six axioms for sets of relations is introduced. All well-known sets of specific orderings, such as linear and weak orderings, satisfy these axioms. These axioms impose criteria of closedness with respect to several operations, such as concatenation, substitution and restriction. For operational reasons and in order to link our results with the literature, it is shown that specific generalizations of the transitivity condition give rise to sets of relations which satisfy these axioms. Next we study minimal extensions of a given set of relations which satisfy the axioms. By this study we come to the fundamentals of orderings: They appear to be special arrangements of several types of disorder. Finally we notice that in this framework many new sets of relations have to be regarded as a set of orderings and that it is not evident how to minimize the number of these new sets of orderings.

KW - Ordering

KW - axiomatization

KW - closure properties

KW - transitivity

U2 - 10.1007/BF00420357

DO - 10.1007/BF00420357

M3 - Article

VL - 9

SP - 367

EP - 396

JO - Order : A Journal on the Theory of Ordered Sets and its Applications

JF - Order : A Journal on the Theory of Ordered Sets and its Applications

SN - 0167-8094

IS - 4

ER -