### Abstract

In this paper a structure of a syslem is defined as a mathematical structure [formula omitted], where [formula omitted]is a first-order logic language and Σ is a set of sentences of the given first-order logic. It is shown that a canonical structure determined by I, which is similar to those used in proving the Gödel's completeness theorem, satisfies a universality in the sense of category theory when homomorphisms are used as morphisms, and a freeness in the sense of universal algebra when Σ-morphisms, which preserve Σ, are used. The universality and the freeness give the minimality of the canonical structure.As an example, a structure of a stationary system is defined as a pair [formula omitted] Its canonical structure is actually constructed. In a sense this canonical structure accords with models constructed by Nerode realization.

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

Pages (from-to) | 141-163 |

Number of pages | 23 |

Journal | International Journal of General Systems |

Volume | 15 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1989 |

Externally published | Yes |

### Fingerprint

### Keywords

- canonical structure
- freeness
- language
- minimality
- model
- stationary system
- Structure
- universality

### ASJC Scopus subject areas

- Computer Science Applications
- Control and Systems Engineering
- Modelling and Simulation
- Theoretical Computer Science
- Information Systems

### Cite this

*International Journal of General Systems*,

*15*(2), 141-163. https://doi.org/10.1080/03081078908935038

**The Canonical Structure as a Minimum Structure.** / Takahara, Yasuhiko; Hjima, Junichi; Takahashi, Shingo.

Research output: Contribution to journal › Article

*International Journal of General Systems*, vol. 15, no. 2, pp. 141-163. https://doi.org/10.1080/03081078908935038

}

TY - JOUR

T1 - The Canonical Structure as a Minimum Structure

AU - Takahara, Yasuhiko

AU - Hjima, Junichi

AU - Takahashi, Shingo

PY - 1989

Y1 - 1989

N2 - In this paper a structure of a syslem is defined as a mathematical structure [formula omitted], where [formula omitted]is a first-order logic language and Σ is a set of sentences of the given first-order logic. It is shown that a canonical structure determined by I, which is similar to those used in proving the Gödel's completeness theorem, satisfies a universality in the sense of category theory when homomorphisms are used as morphisms, and a freeness in the sense of universal algebra when Σ-morphisms, which preserve Σ, are used. The universality and the freeness give the minimality of the canonical structure.As an example, a structure of a stationary system is defined as a pair [formula omitted] Its canonical structure is actually constructed. In a sense this canonical structure accords with models constructed by Nerode realization.

AB - In this paper a structure of a syslem is defined as a mathematical structure [formula omitted], where [formula omitted]is a first-order logic language and Σ is a set of sentences of the given first-order logic. It is shown that a canonical structure determined by I, which is similar to those used in proving the Gödel's completeness theorem, satisfies a universality in the sense of category theory when homomorphisms are used as morphisms, and a freeness in the sense of universal algebra when Σ-morphisms, which preserve Σ, are used. The universality and the freeness give the minimality of the canonical structure.As an example, a structure of a stationary system is defined as a pair [formula omitted] Its canonical structure is actually constructed. In a sense this canonical structure accords with models constructed by Nerode realization.

KW - canonical structure

KW - freeness

KW - language

KW - minimality

KW - model

KW - stationary system

KW - Structure

KW - universality

UR - http://www.scopus.com/inward/record.url?scp=84948876526&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84948876526&partnerID=8YFLogxK

U2 - 10.1080/03081078908935038

DO - 10.1080/03081078908935038

M3 - Article

AN - SCOPUS:84948876526

VL - 15

SP - 141

EP - 163

JO - International Journal of General Systems

JF - International Journal of General Systems

SN - 0308-1079

IS - 2

ER -