E0005

Crystallographic Groupoids: Past, Present, and Future. Carroll K. Johnson, Oak Ridge National Laboratory, Chemical and Analytical Sciences Division, Oak Ridge, TN 37831-6197

Groupoids, and inverse semigroups, are useful for algebraic description of partial and local symmetries which fall outside group theory. Sporadic Crystallographic application of groupoids began shortly after H. Brandt’s groupoid paper in Math. Ann. 96, 360, 1926. Crystallographic examples to be discussed include (a) order-disorder structures; (b) quasicrystals; (c) symmetry of isolated unit cells, asymmetric units, and orbifolds; (d) generalized pseudosymmetry classification; and (e) integrated crystal + surface structure classification.

Our current research centers on item (c). The portion of a standard extended Hermann-Mauguin space group symbol following the lattice symbol P, I, F, etc. describes a screw & glide groupoid (S&GG). The 157 nonsymmo morphic space group symbols contain 122 true S&GGs while the 73 symmo morphic space group symbols contain the 32 point groups. Jaswon and Rose derive and characterize both the space groups and the color groups in a very concise form using S&GGs. We plan to extend their results to the corre sponding crystallographic orbifolds we discussed previously. Items (d) and (e) provide suggestions for future crystallographic research.

(a) K. Dornberger-Schiff and H. Grell, Acta Cryst. A 38, 491-498, 1982;

J. Grell, Acta Applic. Math. 52, 261-269, 1998.

(b) J. Kellendonk and M.V. Lawson, J. Algebra 224, 140-150, 2000.

(c) M.A. Jaswon and M.A. Rose, Crystal Symmetry: Theory of Colour Crystallography, Horwood, 1983.

(d) M.V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries,

World Scientific, 1998.

(e) A. Weinstein, Notices Am. Math. Assoc., July, 1996.