Let I n be the symmetric inverse semigroup on X n = {1, 2, · · · , n} and let DDP n and ODDP n be its subsemigroups of order-decreasing partial isometries and of order-preserving order-decreasing partial isometries of X n , respectively. In this paper we investigate the cycle structure of order-decreasing partial isometry and characterize the Green's relations on DDP n and ODDP n . We show that ODDP n is a 0 − E − unitary ample semigroup. We also investigate the cardinalities of some equivalences on DDP n and ODDP n which lead naturally to obtaining the order of the semigroups. 1 2 MSC2010 : 20M18, 20M20, 05A10, 05A15.
This paper reviews the theory of matrices and determinants. Matrix and determinant are nowadays considered inseparable to some extent, but the determinant was discovered over two centuries before the term matrix was coined. Our review associate determinant with the matrix as part of linear systems but not with polynomials. Thus, the paper first gives the background on matrix with vast applications in all fields of study and then reviews the history of determinants which is based on its major contributors in chronological order from the sixteenth century to the twenty-first century
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