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Parallel transport as dictated by a gauge field determines a collection oflocal reference systems. Comparing local reference systems in overlappingregions leads to an ensemble of algebras of relational kinematical observablesfor gauge theories including general relativity. Using an auxiliary cellulardecomposition, we propose a discretization of the gauge field based on adecimation of the mentioned ensemble of kinematical observables. The outcome isa discrete ensemble of local subalgebras of ``macroscopic observables''characterizing a measuring scale. A set of evaluations of those macroscopicobservables is called an extended lattice gauge field because it determines a$G$-bundle over $M$ (and over submanifolds of $M$ that inherit a cellulardecomposition) together with a lattice gauge field over an embedded lattice. Aphysical observable in our algebra of macroscopic observables is constructed.An initial study of aspects of regularization and coarse graining, which arespecial to this description of gauge fields over a combinatorial base, ispresented. The physical relevance of this extension of ordinary lattice gaugefields is discussed in the context of quantum gravity.

Macroscopic observables from the comparison of local reference systems