Inverse Semigroups and Boolean Matrices,

Abstract

Abstract : Following its fragmentary beginnings in the 1920s and 1930s, the algebraic theory of semigroups has grown from seminal attempts at generalizing group theory into a vast and independent branch of algebra. One subbranch is the extensively developed and exceptionally promising class of inverse semigroups. Intuitively speaking, these semigroups are to partial symmetry what groups are to symmetry. Here we describe software designed to multiply elements of certain inverse semigroups, just as hand calculators multiply numbers. Given the wide range of applications of group theory (symmetry); e.g., understanding roots of polynomials, deriving Laplace spherical functions, understanding rigid-body motion, and classifying quantum particles, it is only natural to consider applications of the more general mathematical theory of partial symmetries. As a first step, the authors have developed software to perform basic (inverse) semigroup operations (multiplications, inverses, etc.). Since the elements of these semigroups may also be pictured as certain matrices of 'Os' and '1s'-usually called monomial or Boolean matrices-the Boolean matrix calculator described in Part 2 is designed to simultaneously display a given semigroup element in both path notation (which exhibits the partial symmetries) and the corresponding monomial ('0-1') matrix. The calculator takes entries in either path notation or matrix notation, and when a Boolean matrix M is the input, the program determines if M represents an element of the semigroup.