Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-27T21:15:01.011Z Has data issue: false hasContentIssue false
  • English
  • Français

The syntax and semantics of entailment in duality theory

Published online by Cambridge University Press:  12 March 2014

B. A. Davey ,
M. Haviar  and
H. A. Priestley
B. A. Davey
Affiliation:
Department of Mathematics, La Trobe University, Bundoora, Victoria 3083, Australia, E-mail: [email protected]
M. Haviar
Affiliation:
Department of Mathematics, M. Bel University, Zvolenska Cesta 6, 974 01 Banska Bystrica, Slovakia, E-mail: [email protected]
H. A. Priestley
Affiliation:
Mathematical Institute, 24/29 St Giles, Oxford OX1 3LB, England, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formula and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of unitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions.

Keywords

Primary 08B9903C9918A40Natural dualitycloneentailmentsyntaxsemantics
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 4 , December 1995 , pp. 1087 - 1114
Copyright
Copyright © Association for Symbolic Logic 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bodnarčuk, V. G.et al., Galois theory for Post algebras, I, II, Kibernetika (Kiev), 1969, no. 3, pp. 1–10; no. 5, pp. 1–9; English translation in Cybernetics, vol. 5 (1969).Google Scholar
[2]Clark, D. M. and Davey, B. A., The quest for strong dualities, Journal of the Austrailian Mathematical Society (to appear).Google Scholar
[3]Clark, D. M. and Davey, B. A., When is a natural duality ‘good’?, Algebra Universalis (to appear).Google Scholar
[4]Clark, D. M. and Davey, B. A., Natural dualities for the working algebraist, Cambridge University Press (in preparation).Google Scholar
[5]Davey, B.A., Duality theory on ten dollars a day, Algebras and orders (Rosenberg, I. G. and Sabidussi, G., editors), NATO Advanced Study Institute Series, Series C, vol. 389, Kluwer, Dordrecht, 1993, pp. 71–111.CrossRefGoogle Scholar
[6]Davey, B. A., Dualisability in general and endodualisability in particular, Proceedings of the International Conference on Logic and Algebra (Siena, April 1994) (to appear).Google Scholar
[7]Davey, B. A., Haviar, M., and Priestley, H. A., Endoprimal distributive lattices are endodualisable, Algebra Universalis (to appear).Google Scholar
[8]Davey, B. A. and Priestley, H. A., Optimal natural dualities, Transactions of the American Mathematical Society, vol. 338 (1993), pp. 655–677.CrossRefGoogle Scholar
[9]Davey, B. A. and Priestley, H. A., Natural dualities. II: general theory, Translations of the American Mathematical Society (to appear).Google Scholar
[10]Davey, B. A. and Werner, H., Dualities and equivalences for varieties of algebras, Contributions to lattice theory (Szeged, 1980), (Huhn, A.P. and Schmidt, E.T., eds) Colloqeria Mathematica Societalis János Bolyai, vol. 33, North–Holland, Amsterdam, 1983, pp. 101–275.Google Scholar
[11]Jónsson, B., Algebras whose congruence lattices are distributive, Mathematica Scandinavica, vol. 21 (1967), pp. 110–121.CrossRefGoogle Scholar
[12]Pixley, A. F., Semi-categorical algebras. II, Mathematische Zeitschrift, vol. 85 (1964), pp. 169–184.Google Scholar
[13]Pöschel, R. and Kalužnin, L. A., Relationenalgebren, VEB Deutscher Verlag der Wissenschaften, Berlin, 1979.CrossRefGoogle Scholar
[14]Zadori, L., Natural duality via a finite set of relations, Bulletin of the Australian Mathematical Society (to appear).Google Scholar