Geodesic
Residuation theory and matrix multiplication on orthomodular lattices
Glasgow mathematical journal, Tome 10 (1969) no. 1, pp. 60-65

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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 a ≦ b ⇒ a ∨ (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(Aik ∧ BkJ). 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 X → XP (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. 
Bevis, J. H.; Martin, C. K. Residuation theory and matrix multiplication on orthomodular lattices. Glasgow mathematical journal, Tome 10 (1969) no. 1, pp. 60-65. doi: 10.1017/S0017089500000549
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