Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T11:53:41.372Z Has data issue: false hasContentIssue false
  • English
  • Français

Some large classes of n-valued Sheffer functions

Published online by Cambridge University Press:  24 October 2008

J. C. Muzio
J. C. Muzio
Affiliation:
University of Manitoba, Winnipeg, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

In 1967, Singer (11) gave 3 classes of n-valued two-place functors and proved that all these functors were Sheffer functions. Out of the n possible assignments needed to define a functor completely, Singer showed that it was sufficient to define 3n − 2, 3n − 2, and 2n assignments respectivelyfor the 3 classes. We shall enlarge Singer's classes to give functors of type Ia, type II and type III. For types Ia and III, it will be shown that it is sufficient to define 2n − 1 assignments and for type II we require 2n − 1 assignments to be defined and conditions on a further n/p1 assignments (where P1 is the least prime factor of n). These classes of functors include all of Singer's classes. We also introduce functors of type Ib, similar to those of type Ia, and show that for these itis sufficient to define 2n − 1 assignments to ensure the functor is a Sheffer function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 2 , September 1973 , pp. 201 - 211
Copyright
Copyright © Cambridge Philosophical Society 1973

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)Davies, R. O.On n-valued Sheffer functions. J. Symbolic Logic, forthcoming.Google Scholar
(2)Evans, T. and Hardy, L.Sheffer stroke functions in many valued logics. Portugal Math. 16 (1957), 83–93.Google Scholar
(3)Graham, R. L.On n-valued functionally complete truth functions. J. Symbolic Logic 32 (1967), 190–195.CrossRefGoogle Scholar
(4)Martin, N. M.Some analogues of the Sheffer stroke function in n-valued logic. Indag. Math. 12 (1950), 393–400.Google Scholar
(5)Martin, N. M.The Sheffer functions of 3-valued logic. J. Symbolic Logic 19 (1954), 45–51.Google Scholar
(6)Muzio, J. C.On the t-closure condition of Martin. Math. Ann. 195 (1972), 143–148.CrossRefGoogle Scholar
(7)Rosenberg, I.La structure des fonctions de plusieurs variables sur un ensemble fini. C.R. Acad. Sci. Paris Sér. A.B 260 (1965), 3817–3819.Google Scholar
(8)Rousseau, G.Completeness in finite algebras with a single operation. Proc. Amer. Math. Soc. 18 (1967), 1009–1013.Google Scholar
(9)Salomaa, A.On the composition of functions of several variables rangingover a finite set. Ann. Univ. Turku, 5cr AI 41 (1960).Google Scholar
(10)Schofield, P.Independent conditions for completeness of finite algebras with a single generator. J. London Math. Soc. 44 (1969), 413–423.CrossRefGoogle Scholar
(11)Singer, F. R.Some Sheffer functions for n-valued logics. Scripta Math. 28 (1967), 21–27.Google Scholar
(12)Webb, D. L.Generation of any n-valued logic by one binary operation. Proc. Nat. Acad. Sin. (U.S.A.) 21 (1935), 252–254.Google Scholar
(13)Wheeler, R. F.Complete connectives for the 3-valued propositional calculus. Proc. London Math. Soc. 16 (1966), 167–191.CrossRefGoogle Scholar
(14)Yablonskii, S. V.Functional constructions in k-valued logic (in Russian). Trudy Mat. hut. Steklov 51 (1958), 5–142.Google Scholar