On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions

Hans-Jürgen Vogel

Discussiones Mathematicae - General Algebra and Applications (2001)

  • Volume: 21, Issue: 2, page 139-163
  • ISSN: 1509-9415

Abstract

top
The category of all binary relations between arbitrary sets turns out to be a certain symmetric monoidal category Rel with an additional structure characterized by a family d = ( d A : A → A ⨂ A | A ∈ | R e l | ) of diagonal morphisms, a family t = ( t A : A → I | A ∈ | R e l | ) of terminal morphisms, and a family ∇ = ( ∇ A : A ⨂ A → A | A ∈ | R e l | ) of diagonal inversions having certain properties. Using this properties in [11] was given a system of axioms which characterizes the abstract concept of a halfdiagonal-halfterminal-symmetric monoidal category with diagonal inversions (hdht∇s-category). Besides of certain identities this system of axioms contains two identical implications. In this paper is shown that there is an equivalent characterizing system of axioms for hdht∇s-categories consisting of identities only. Therefore, the class of all small hdht∇-symmetric categories (interpreted as hetrogeneous algebras of a certain type) forms a variety and hence there are free theories for relational structures.

How to cite

top

Hans-Jürgen Vogel. "On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions." Discussiones Mathematicae - General Algebra and Applications 21.2 (2001): 139-163. <http://eudml.org/doc/287740>.

@article{Hans2001,
abstract = {The category of all binary relations between arbitrary sets turns out to be a certain symmetric monoidal category Rel with an additional structure characterized by a family $d = (d_\{A\}: A → A⨂ A | A ∈ |Rel|)$ of diagonal morphisms, a family $t = (t_\{A\}: A → I | A ∈ |Rel|)$ of terminal morphisms, and a family $∇ = (∇_\{A\}: A ⨂ A → A | A ∈ |Rel|)$ of diagonal inversions having certain properties. Using this properties in [11] was given a system of axioms which characterizes the abstract concept of a halfdiagonal-halfterminal-symmetric monoidal category with diagonal inversions (hdht∇s-category). Besides of certain identities this system of axioms contains two identical implications. In this paper is shown that there is an equivalent characterizing system of axioms for hdht∇s-categories consisting of identities only. Therefore, the class of all small hdht∇-symmetric categories (interpreted as hetrogeneous algebras of a certain type) forms a variety and hence there are free theories for relational structures.},
author = {Hans-Jürgen Vogel},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {halfdiagonal-halfterminal-symmetric category; diagonal inversion; partial order relation; subidentity; equation; halfdiagonal-halfterminal-symmetric monoidal category; diagonal inversions},
language = {eng},
number = {2},
pages = {139-163},
title = {On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions},
url = {http://eudml.org/doc/287740},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Hans-Jürgen Vogel
TI - On the structure of halfdiagonal-halfterminal-symmetric categories with diagonal inversions
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2001
VL - 21
IS - 2
SP - 139
EP - 163
AB - The category of all binary relations between arbitrary sets turns out to be a certain symmetric monoidal category Rel with an additional structure characterized by a family $d = (d_{A}: A → A⨂ A | A ∈ |Rel|)$ of diagonal morphisms, a family $t = (t_{A}: A → I | A ∈ |Rel|)$ of terminal morphisms, and a family $∇ = (∇_{A}: A ⨂ A → A | A ∈ |Rel|)$ of diagonal inversions having certain properties. Using this properties in [11] was given a system of axioms which characterizes the abstract concept of a halfdiagonal-halfterminal-symmetric monoidal category with diagonal inversions (hdht∇s-category). Besides of certain identities this system of axioms contains two identical implications. In this paper is shown that there is an equivalent characterizing system of axioms for hdht∇s-categories consisting of identities only. Therefore, the class of all small hdht∇-symmetric categories (interpreted as hetrogeneous algebras of a certain type) forms a variety and hence there are free theories for relational structures.
LA - eng
KW - halfdiagonal-halfterminal-symmetric category; diagonal inversion; partial order relation; subidentity; equation; halfdiagonal-halfterminal-symmetric monoidal category; diagonal inversions
UR - http://eudml.org/doc/287740
ER -

References

top
  1. [1] L. B and H.-J. H, Automaten und Funktoren, Akademie-Verlag, Berlin 1975. 
  2. [2] S. E and G.M. K, Closed categories, 'Proceedings of the Conference on Categorical Algebra (La Jolla, 1965)', Springer-Verlag, New York 1966, 421-562. 
  3. [3] H.-J. H, On partial algebras, Colloquia Mathematica Societatis János Bolyai, 29 ('Universal Algebra, Esztergom (Hungary) 1977'), North-Holland, Amsterdam 1981, 373-412. 
  4. [4] G.M. K, On MacLane's condition for coherence of natural associativities, commutativities, etc, J. Algebra 1 (1964), 397-402. Zbl0246.18008
  5. [5] S. M, Kategorien, Begriffssprache und mathematische Theorie, Springer-Verlag, Berlin-New York 1972. 
  6. [6] J. S, Über dht-symmetrische Kategorien, Semesterarbeit Päd. Hochschule Köthen, Köthen 1978. 
  7. [7] J. S, Über die Einbettung von dht-symmetrischen Kategorien in die Kategorie der partiellen Abbildungen zwischen Mengen, AdW DDR Berlin, Zentralinstitut f. Mathematik und Mechanik, Preprint P-12/8 (1980). Zbl0442.18002
  8. [8] J. S, Zur Theorie der dht-symmetrischen Kategorien, Dissertation (B), eingereicht an der Math.-Naturwiss. Fak. d. Wiss. Rates d. Pädag. Hochschule 'Karl Liebknecht', Potsdam 1984. 
  9. [9] J. S, Zur Axiomatik von Kategorien partieller Morphismen, Beiträge Algebra Geom. (Halle/Saale) 24 (1987), 83-98. 
  10. [10] H.-J. V, Eine kategorientheoretische Sprache zur Beschreibung von Birkhoff-Algebren, AdW DDR, Institut f. Mathematik, Report R-MATH-06/84, Berlin (1984). 
  11. [11] H.-J. V, Relations as morphisms of a certain monoidal category, 'General Algebra and Applications in Discrete Mathematics', Shaker Verlag, Aachen 1997, 205-217. 
  12. [12] H.-J. V, On functors between dhtŃ-symmetric categories, Discuss. Math. - Algebra & Stochastic Methods 18 (1998), 131-147. Zbl0921.18005
  13. [13] H.-J. V, On Properties of dhtŃ-symmetric categories, Contributions to General Algebra 11 (1999), 211-223. 
  14. [14] H.-J. V, Halfdiagonal-halfterminal-symmetric monoidal categories with inversions, 'General Algebra and Discrete Mathematics', Shaker Verlag, Aachen 1999, 189-204. 
  15. [15] H.-J. V, On morphisms between prtial algebras, 'Algebras and Combinatorics. An International Congress, ICAC '97, Hong Kong', Springer-Verlag, Singapore 1999, 427-453. Zbl0968.08003

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.