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Residuation theory and Boolean matrices

Published online by Cambridge University Press:  18 May 2009

T. S. Blyth
T. S. Blyth
Affiliation:
St Salvator's CollegeSt Andrews
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We begin this paper by considering a Boolean algebra as a lattice which is relatively pseudo-complemented (i.e., residuated with respect to intersection) and give, in this case, certain properties of the equivalences of types A, B and F(as introduced by Molinaro [1]). We then show how these results carry over to the case of Boolean matrices, which form a Boolean algebra residuated also with respect to matrix multiplication. Other properties of matrix residuals are established and we conclude with three algebraic characterisations of invertible Boolean matrices.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 6 , Issue 4 , July 1964 , pp. 185 - 190
Copyright
Copyright © Glasgow Mathematical Journal Trust 1964

References

REFERENCES

1.Molinaro, I., Demi-groupes résidutifs, D. ès Sc. thesis, Paris, 1956; also J. Math. Pures Appl. 39 (1960), 319–356 and 40 (1961), 43–110.Google Scholar
2.Luce, R. D., A note on Boolean matrix theory, Proc. Amer. Math. Soc. 3 (1952), 382–388.CrossRefGoogle Scholar
3.Rutherford, D. E., Inverses of Boolean matrices, Proc. Glasgow Math. Assoc. 6 (1963), 49–53.CrossRefGoogle Scholar