is called the generalized inverse or Moore–Penrose inverse. Semirings Examples All examples in this section involve associative operators. Galois connections The lower and upper adjoints in a (monotone) Galois connection, L and G are quasi-inverses of each other; that is, LGL = L and GLG = G and one uniquely determines the other. They are not left or right inverses of each other however. Generalized inverses of matrices A square matrix with entries in a field is invertible (in the set of all square matrices of the same size, under matrix multiplication) if and only if its determinant is different from zero. If the determinant of is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one. See invertible matrix for more. More generally, a square matrix over a commutative ring is invertible if and only if its determinant is invertible in . Non-square matrices of full rank have several one-sided inverses: For we have left inverses; for example, For we have right inverses; for example, The left inverse can be used to determine the least norm solution of , which is also the least squares formula for regression and is given by No rank deficient matrix has any (even one-sided) inverse. However, the Moore–Penrose inverse exists for all matrices, and coincides with the left or right (or true) inverse when it exists. As an example of matrix inverses, consider: So, as m < n, we have a right inverse, By components it is computed as The left inverse doesn't exist, because which is a singular matrix, and cannot be inverted. See also Division ring Latin square property Loop (algebra) Unit (ring theory) Notes References M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, , p. 15 (def in unital magma) and p. 33 (def in semigroup) contains all of the semigroup material herein except *-regular semigroups. Drazin, M.P., Regular semigroups with involution, Proc. Symp. on Regular Semigroups (DeKalb, 1979), 29–46 Miyuki Yamada, P-systems in regular semigroups, Semigroup Forum, 24(1), December 1982, pp. 173–187 Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups, Semigroup Forum, 16(1978), 369–377. Algebra Abstract algebra Inverse element