Tight upper bounds for boolean operations on right ideals [8] are mn for intersection and symmetric difference, mn - (m -1) for difference, and mn - (m + n - 2) for union.

Symmetric Difference, Difference, and Union: The arguments are similar to those used in the proof of Theorem 4.

We list here each operation with the size of the smallest known alphabet (first entry) along with our alphabet size (second entry): reversal (2/2), star (2/2), product (2/3), union (2/3), intersection (2/3), symmetric difference (2/3), and difference (2/3).

The argument is symmetric to that for Claim 1, when the operation is intersection, symmetric difference, or union.

The sizes are as follows: reversal (3/4), star (2/2), product (1/2), union (4/3), intersection (2/3), symmetric difference (2/3), and difference (3/3).

For symmetric difference, every state in Row m or Column n except (m', n) is final, and (m', n) is the only empty state.

Given any graph G, letM(G) be the graph whose vertex set is the set of perfect matchings of G, where two vertices of M(G) are adjacent whenever the symmetric difference of the corresponding perfect matchings is a cycle of length 4.

M'(G) has as its vertex set the perfect matchings of G, where two matchings are now adjacent if and only if their symmetric difference is a cycle of arbitrary length.

i) In the geometric version of this graph, the vertices are the non-crossing perfect matchings of a set of 2n points in convex position; two of which are adjacent whenever their symmetric difference is a cycle of length four.

Subjects include non-normal distributions that describe the Bayesian updating of atmospheric models, efficient uniform designs for mixture experiments in three and four components, designs of accelerated life tests for periodic inspection with Burr Type III distributions, parameter estimation using Cressie-Read divergence measures with exponential grouped censored data, estimation of variance components of acceleration degradation models, production of the ration of the

symmetric differences of order statistics, surface roughness measurements acquired by spatial statistics, Diallel crosses, and the characterization of distributions by conditional expectations of functions of generalized order statistics.