June 1, 2007: Friday, 1:30-3:00 telecast from UCLA 285 Powell Library

Michael Merrill, Arizona State University

School of Human Evolution & Social Change

"Archaeology and Galois lattices"

http://eclectic.ss.uci.edu/~drwhite/center/cac.html#Merrill

Detailed Abstract: I will be analyzing structural patterning in the attributes of artifacts called Megathrua crenulata (Giant Keyhole Limpet) shell ornaments. The ornaments were recovered from ten burials (providing a 10% sample size and excellent spatial coverage of a cemetery that was nearly completely excavated by David Banks Rogers between April 10 and June 28, 1926) in a mainland coastal Chumash site (SBa-72 south) that was occupied from approximately A.D. 900 to 1150. The impetus for the analysis is to find a way to explore further Chester King's comment that "the size of the shell determines the size of the callus ring in the center of the shell" (King 1990:125), which in turn determines the economic value of the shell as well as the exchange value, social context and use of the finished ornament.

Two primary techniques, chipping and grinding, were used to reduce and shape the shell area surrounding the callus ring. Grinding requires a greater input of time and energy and results in a more nicely shaped and presumably a more valuable finished product. Also, larger shells have more material that can be removed, which suggests a selection bias for larger shells in ornament types, such as rings, that require a maximum removal of material to make a usable ornament. The two main hypotheses being tested in this analysis are (1) that the energetics of manufacture strongly relates to the selection of shell size and (2) that emic size categories (categories recognized and selected for by the makers and users of these ornaments) provide structure for the data. A third hypothesis being tested is that there is a strong temporal component to the callus ring size class and ornament type dependencies that relates to the presence of already established temporal diagnostic forms, specifically Olivella split punched and cupped beads, that were found within, or in close proximity to, a subset of the ten burials in the sample. I will provide photographs of the different types of Megathura ornaments and Olivella beads as part of my presentation.

My analytical method is based on determining structure through representing the artifact data with Galois lattices. Galois lattices are dually ordered algebraic structures that provide a very robust and elegant conceptual framework for exploring structural patterning in archaeological data. The Galois lattice has a formally defined structure that incorporates set-theoretic principles such as set union and overlap in conjunction with other mathematical concepts that include least upper bound (supremum) and greatest lower bound (infimum), which are used to assemble a partially ordered set (the lattice). A Galois lattice is most often visualized in terms of a diagram (often called a Hasse diagram) consisting of nodes (lattice elements) and lines connecting the nodes.

Each lattice node in this analysis is associated with one or more shell ornaments from a burial and/or one shell ornament type. The nodes of the computer generated lattice diagrams in my talk will be labeled accordingly. A specific property of Galois lattices known as the Luxenburger basis (Luxenburger 1991) will be used to uncover sets of absolute and partial (true most but not all of the time) implications or dependencies (also known as "association rules") between size classes of the siphon hole or callus ring of Megathrua crenulata (Giant Keyhole Limpet) ornaments and specific types (e.g. wing-shaped rectangular) of these ornaments The detailed structure in the data determined through using Galois lattices would not be revealed by any of the multivariate methods used by archaeologists such as principal components analysis and correspondence analysis. I will discuss the methodology employed in the analysis in a step-by-step manner, along with providing needed information about Galois lattices and the Luxenburger basis.

References

King, Chester D. 1990. Evolution of Chumash Society. A Comparative Study of Artifacts Used for Social System Maintenance in the Santa Barbara Channel Region before A.D. 1804. Garland Publishing, Inc. New York.

Luxenburger, Michael. 1991. Implications Partielles Dans Un Contexte. Math. Inf. Sci. hum. 29(113): 35-55.