Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T03:09:01.105Z Has data issue: false hasContentIssue false

Abstract orthogonality and orthocomplementation

Published online by Cambridge University Press:  24 October 2008

Gianpiero Cattaneo
Affiliation:
Istituto di Scienze Fisiche dell'università, 20133 Milano, Italy
Alessandro Manià
Affiliation:
Istituto di Scienze Fisiche dell'università, 20133 Milano, Italy

Abstract

The notion of orthogonality is axiomatically defined on a poset. Various notions of orthocomplementation are distinguished and conditions are given in order to induce an orthocomplementation from an orthogonality and vice versa. Subsequently ⊥-modular ⊥-poset are defined and the set of morphisms between two posets with orthogonality is briefly discussed. Given the notions of additive monoid and of positive semi-ring, an orthogonality relation is introduced on the set of idempotent elements of a positive semi-ring. Finally, the obtained results are applied to the set of idempotent and absorbent endomorphisms of an additive monoid.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1974

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)Lorenzen, P.Metamathematik, B-1-Hochschultaschenbücher, 25 (1962), Mannheim.Google Scholar
(2)Varadarajan, V. S.(i) Probability in physics and a theorem on simultaneous observability. Comm. Pure Appl. Math. XV, no. 2, 189216 (1962). (ii) Geometry of quantum theory, vol. I (Van Nostrand, 1970).CrossRefGoogle Scholar
(3)Birkhoff, G. and Von Neumann, J.The logic of quantum mechanics. Ann. of Math. 37 (1936), 823843.CrossRefGoogle Scholar
(4)Mielnik, B.(i) Geometry of quantum states. Comm. Math. Phys. 9 (1968), 5580. (ii) Theory of filters. Comm. Math. Phys. 15 (1969), 1–46.CrossRefGoogle Scholar
(5)Pool, J.Baer*-semigroups and the logic of quantum mechanics. Comm. Math. Phys. 9 (1968), 118141.CrossRefGoogle Scholar
(6)Birkhoff, G.Lattice theory (American Mathematical Society; Providence, 1967).Google Scholar
(7)Cattaneo, G. and Mani′, A. General information systems in empirical sciences, preprint IFUM 148/FT, 04 1973.Google Scholar
(8)Jauch, J.Foundations of Quantum Mechanics (Addison-Wesley; Reading, Mass., 1968).Google Scholar