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Residuation theory and matrix multiplication on orthomodular lattices

  • J. H. Bevis (a1) and C. K. Martin (a1)

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In this paper we consider mappings induced by matrix multiplication which are defined on lattices of matrices whose coordinates come from a fixed orthomodular lattice L (i.e. a lattice with an orthocomplementation denoted by ′ in which aba ∨ (a′b) = b). will denote the set of all m × n matrices over L with partial order and lattice operations defined coordinatewise. For conformal matrices A and B the (i,j)th coordinate of the matrix product AB is defined to be (AB)ij = Vk(AikBkJ). We assume familiarity with the notation and results of [1]. is an orthomodular lattice and the (lattice) centre of is defined as , where we say that A commutes with B and write . In § 1 it is shown that mappings from into characterized by right multiplication XXP (P) are residuated if and only if p ∈ ℘ (). (Similarly for left multiplication.) This result is used to show the existence of residuated pairs. Hence, in § 2 we are able to extend a result of Blyth [3] which relates invertible and cancellable matrices (see Theorem 3 and its corollaries). Finally, for right (left) multiplication mappings, characterizations are given in § 3 for closure operators, quantifiers, range closed mappings, and Sasaki projections.

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References

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1.Bevis, J. H., Matrices over orthomodular lattices, Glasgow Math. J. 10 (1968), 5559.
2.Birkhoff, G., Lattice theory, Amer. Math. Soc. Colloquium Publications, Vol. 25, rev. ed. (New York, 1948).
3.Blyth, T. S., Residuation theory and Boolean matrices, Proc. Glasgow Math. Assoc. 6 (1964), 185190.
4.Croisot, R., Applications residuées, Ann. Sci. Ecole Norm. Sup. (3) 73 (1956), 453474.
5.Foulis, D. J., Baer *-semigroups, Proc. Amer. Math. Soc. 11 (1960), 648654.
6.Foulis, D. J., Conditions for the modularity of an orthomodular lattice, Pacific J. Math. 11 (1961), 889895.
7.Janowitz, M. F., Quantifiers and orthomodular lattices, Pacific J. Math. 13 (1963), 12411249.
8.Janowitz, M. F., A semigroup approach to lattices, Canad.J. Math. 18 (1966), 12121223.
9.Molinaro, I., Demi-groupes résidutifs, J. Math. Pures Appl. 39 (1960), 319356.
10.Rutherford, D. E., Inverses of Boolean matrices, Proc. Glasgow Math. Assoc. 6 (1963), 4953

Residuation theory and matrix multiplication on orthomodular lattices

  • J. H. Bevis (a1) and C. K. Martin (a1)

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